Behaviour and properties of natural soils and soft rocks

S. LeroueilDepartment of Civil Engineering, Université Laval, Ste-Foy, Québec, Canada, G1K 7P4

D.W. HightGeotechnical Consulting Group, 1a Queensberry Place, London, United Kingdom, SW7 2DL

ABSTRACT: This paper is an overview of the mechanical and hydraulic behaviour and properties ofnatural soils and soft rocks, from soft clays to chalk. It is based on Critical State Soil Mechanics, but theassociated concepts are extended to take into account the influence of the major features of natural soilsthat are possible: anisotropy, microstructure, viscosity, partial saturation and discontinuities. The firstpart of the paper deals with soil behaviour (14 sections); the second part is more oriented towards soilproperties (6 sections).

1 INTRODUCTION

Classical soil mechanics developed mainly in the first half of the twentieth century. In particular, the fol-lowing developments can be mentioned: consolidation theory (Terzaghi, 1923), concept of effective stress(Terzaghi, 1936), concept of preconsolidation pressure (Casagrande, 1936b), understanding of strengthcomponents (Hvorslev, 1937), description of soil compaction and consistency (Proctor, 1933; Atterberg,1911), and understanding of drained vs undrained soil behaviour (Bishop; Skempton). These works stillcover about 70% of our soil mechanics textbooks. What can be called Modern Soil Mechanics was born atthe turn of the nineteen-sixties, in particular with Roscoe, Schofield, Wroth and co-workers who integratedshear stress, mean effective stress, void ratio and shear strain in the same framework, often referred to asCritical State Soil Mechanics (CSSM). This work was reinforced with the stress path method (Lambe, 1967)and the concept of normalized soil behaviour (Ladd, 1969). It is also at that time that the influence of dila-tancy was fully understood (Rowe, 1962; Lee & Seed, 1967). It was however in the nineteen-eighties thatsome major features of natural soils have been fully recognised: small strain stiffness (Jardine, 1985;Burland, 1989) and its measurement by geophysical methods (Stokoe & Nazarian, 1985; Stokoe et al.,1991); concept of collapse surface for explaining static liquefaction of sand (Sladen et al., 1985); importanceof effect of strain rate (Graham et al., 1983a); influence of anisotropy and stress axis rotation (Hight and co-workers); extension of CSSM concepts to unsaturated soils (Alonso et al., 1987, 1990); the fact that mostnatural geomaterials are microstructured (Burland, 1990; Leroueil & Vaughan, 1990). In the last fifteenyears or so: (a) these ideas have essentially been verified for a variety of geomaterials; (b) equipment, in par-ticular for small strain and shear wave measurement, has been improved; and (c) more and more powerfulnumerical models integrating recent findings have really started being applied to analyses in major projects.

The aim of this paper is to synthesize this Modern Soil Mechanics into a general framework thatcould be applied to a variety of geomaterials, from soft clays to soft rocks, considered homogeneous.The basic concepts of CSSM are used as the foundation of this framework. Then, major features of nat-ural soil behaviour, such as anisotropy, microstructure, viscosity and partial saturation are included inthe framework as extensions of the basic CSSM. The preparation of the paper has been greatly facili-tated by the reading of contributions to this Workshop.

The paper contains 19 sections in addition to the introduction and a brief conclusion. The sections aredivided in two parts: Part I deals mainly with soil behaviour; and Part II is more oriented towards soilproperties. The general outline is as follows:

Part I: Soil behaviour– Basic elasto-plastic model and failure criteria for soils (Section 2)– Limit state curves of natural soils and soft rocks (Section 3)

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Characterisation and Engineering Properties of Natural Soils – Tan et al. (eds.)© 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 537 1

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– Behaviour inside the limit state curve (Section 4)– Behaviour of cohesionless soils (Section 5)– Influence of anisotropy (Section 6)– Behaviour of intermediate soils (Section 7)– Post-failure behaviour (Section 8)– Influence of strain rate and time (Section 9)– Influence of temperature (Section 10)– Influence of microstructure (Section 11)– Influence of discontinuities (Section 12)– Influence of soil water chemistry (Section 13)– Influence of partial saturation (Section 14)

Part II: Soil properties– Generalities on soil properties (Section 15)– Properties of soils in remoulded conditions (Section 16)– Mechanical properties of intact soil (Section 17)– Properties associated with the passage from intact to destructured, possibly remoulded, conditions

(Section 18)– Hydraulic properties (Section 19)– Characteristics of consolidation (Section 20)

Some parts of the paper will appear trivial to many readers. This has (partly) been done on purpose,the paper aiming at being both close to a state-of-the-art paper and a tool that could be used for pedago-logical purposes. It is, indeed, an extended and updated version of the “Advanced Soil Mechanics”course available at Université Laval, Québec.

2 BASIC ELASTO-PLASTIC MODEL (CRITICAL STATE SOIL MECHANICS) AND FAILURE CRITERIA FOR SOILS

2.1 Elasto-plastic behaviour of ideal soils (Critical state soil mechanics)

Basic concepts of Critical State Soil Mechanics (CSSM) were established in the late nineteen-fiftiesand -sixties in England (Roscoe et al., 1958; Schofield & Wroth, 1968), mostly on the basis of tests per-formed on reconstituted and isotropically consolidated samples of clay. It is representative of saturatedsoils which are assumed to be isotropic and not influenced by factors such as strain rate and microstruc-ture. CSSM thus describes the behaviour of an “ideal soil” and can be summarized as below.

When a clayey soil, initially at a water content close to its liquid limit, is loaded isotropically, its voidratio decreases along the normal compression line (NCL), a straight line in an e–log p� (or v–ln p�) dia-gram1 (Figure 2.1a). If after being loaded to p�A, at point A, the soil is progressively unloaded isotropi-cally, there is a slight rebound along AR, also a straight line in the e–log p� diagram. If reloaded fromR, the soil first behaves elastically from R to A, and then follows the normal compression line along AB,where deformations are mostly plastic. For the soil at R, A is a yield point or a limit state, at which itstarts developing plastic strains. The NCL and the recompression line are characterized by their slopes, � �e/�ln p� (or the compression index Cc � �e/�log p�) and � �e/�ln p�.

In a p�–q diagram, a soil loaded at point A, under an isotropic effective stress p�A, generates a stressdomain in which soil behaviour is elastic. This domain is limited by the yield or limit state curve pass-ing through point A (Figure 2.1b). For a stress path such as LM, inside the limit state curve, the behav-iour is elastic; whereas for a stress path such as LT, it is elastic from L to S, and plastic strain willdevelop from S to T. For the soil loaded to point B (Figure 2.1a), the limit state curve becomes thedashed line passing through B in Figure 2.1b. The new limit state curve is associated with the accumu-lated volumetric plastic strain, and its shape is the same (isotropic hardening), regardless of the stresspath along which it has been created. In an (e–p�–q) diagram (Figure 2.2b) the normal compression lineand the limit state curves generate the limit state (or state boundary) surface. When stress conditions onthe limit state surface are normalized with respect to the mean effective stress on the NCL at the same void ratio, p�e, then they lie on the same line in a normalized p�/p�e–q/p�e diagram(Figure 2.3).

30

1Definition of symbols is given in Section 22, at the end of the text.

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When yielding occurs at a stress state such as S in Figures 2.1b and 2.4, it is associated with thedevelopment of plastic strains. The plastic strain increments can be described by ��v

p and ��sp which can

be plotted at point S to form a plastic strain increment vector SV. �v (also termed �vol) is the volumetricstrain (�V/Vo � �1 � �2 � �3) and �s is the shear strain (2(�1 � �1)/3) in axisymmetic conditions),

31

e

CSL

q(a) (b)CSL

Normal compression line (NCL)

RA

B

Yield or limit state curve

M

L

T

S

A B

Log p´

P´

Figure 2.1. Limit state or yield or state boundary surface.

P´

P´

q

q

e

C

CC

T

TT

N N NO

T

T

TN

N

N

C

C

C

NCL

CSL

Hvorslev surface

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3

123

q = Mp´

(a)

(b)

Figure 2.2. Limit state or yield or state boundary surface.

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with �1, �2 and �3 being the major, the intermediate and minor principal strains. It is generally assumedthat vector SV is perpendicular to the yield or limit state curve; it is then said that there is normality orassociated flow.

At large strains in shear tests, the soil tends towards an ultimate condition at which there is no changein void ratio and effective stresses. This condition is called critical state or fully softened state. Criticalstate can be obtained by drained and undrained shear tests performed on normally or slightly overcon-solidated clay samples; it is then reached at large strain at the same time as the maximum deviatoric,( 1 – 3), or shear stress. For a heavily overconsolidated soil, critical state is also reached at large strain,but after a peak and strain softening in the case of drained shear tests.

Critical state conditions are on a curve, (CSL), in the e–p�–q diagram (Figure 2.2). Projected onto theq–p� space, the CSL becomes straight and is defined as follows:

qcs � Mp�cs (2.1)

32

q/p e

C

A

M

M

p/pe

Ext

ensi

onC

ompr

essi

on

c

e

Hvorslevsurface

Limit statesurface

Critical state

0.5 1.0O

´

´ ´

Figure 2.3. Limit state surface and Hvorslev surface in both compression and extension, in the normalizedp�/p�e–q/p�e diagram.

P´

CSLq

L

T

S

A

V

pv

ps

pv

p sδε

δε

δε

δε

Figure 2.4. Limit state curve and plastic strain increment vector when normality applies.

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where: M � Mc � 6 sin ��/(3 � sin ��) for triaxial compression conditions and M � Me � 6 sin ��/(3 � sin ��) for triaxial extension conditions.

Equation 2.1 is also the Mohr-Coulomb criterion defined for normally consolidated materials and � �is the corresponding friction angle, ��nc, or critical state friction angle ��cs. Typical values of frictionangles and relationships with other soil parameters are presented in Section 16.5.

In the e–log p� diagram, the projection of CSL is parallel to the normal compression line(Figure 2.1a). Consequently, in a p�/p�e–q/p�e diagram (Figure 2.3), CSL is represented by a uniquepoint, C.

On the basis of tests performed with the direct shear box on normally and overconsolidated speci-mens of clay, Hvorslev (1937) came to the conclusion that the peak shear strength �f under an effectivestress � can be described as follows:

�f � c�e � � tan ��e (2.2)

where ��e is the Hvorslev angle of shearing resistance and c�e is the Hvorslev cohesion, which increaseswhen the void ratio decreases.

As shown by Schofield & Wroth (1968) and by Wood (1990),

c�e � c�pe p�e (2.3)

c�pe is the Hvorslev cohesion parameter. Thus the strength envelopes can be described as follows in theq/p�e–p�/p�e diagram:

– In triaxial compression

(2.4)

with

(2.5)

– In triaxial extension

(2.6)

with

(2.7)

These normalized strength envelopes are shown in Figure 2.3. In the e–p�–q diagram, they generatethe upper (and lower) part of the limit state surface, also called the Hvorslev surface (Figure 2.2b).

The previously mentioned elements constitute the basics of CSSM that apply well to “ideal” soils.They are essential to the understanding of soil behaviour and constitute the foundation on which it ispossible to build a modern soil mechanics to take into account the features of “real” soils. These features will be examined in the following sections.

A simple application of CSSM to triaxial compression tests is illustrated in Figure 2.5 for soil spec-imens normally consolidated at point M and then isotropically unloaded at different overconsolidationratios, OCRs, before shearing. OCR is defined as the ratio of the maximum axial or mean stress appliedin the past over the currently applied stress. The soil, which initially is in the overconsolidated rangebehaves elastically until it reaches the limit state curve (at points A1, B1, C1 and D1) and then progres-sively moves towards the critical state corresponding either to the effective stress path followed indrained tests (AA1A2 and BB1B2) or its void ratio in undrained tests (AC1C2 and BD1D2). It can be

M6 sin

sineee

e�

�

� �

′′3

q

p

6 cos

sinc M

p

pe

e

epe ee

e′′′

′′′

� ��

� ��

3

M6 sin

sincee

e�

�

� �

′′3

q

p

6 cos

sinc M

p

pe

e

epe ce

e′′′

′′′

��

� ��

3

33

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noted, in particular, that the drained test on the slightly overconsolidated soil specimen (stress pathAA1A2) shows ductile behaviour whereas the drained test on the strongly overconsolidated soil speci-men (stress path BB1B2) shows a strain softening behaviour.

2.2 Failure criteria

It is generally accepted that failure of soils is controlled by the Mohr-Coulomb criterion

�f � c� � � tan �� (2.8)

In an octahedral (p� � cst) plane, this criterion is an irregular hexagon, as shown in Figure 2.6.Experience however shows that the failure surface is more rounded than indicated by the Mohr-Coulomb criterion.

Matsuoka & Nakai (1974) and Lade & Duncan (1975) proposed slightly different criteria that aredescribed as follows:

1) Matsuoka & Nakai criterion:

(2.9)I I

Icst1 2

3

⋅�

34

e or

w P'

B

BBD

D

A AM

A

CC

NCL

CSL

12

1

2

12

1

2

A2

C2

D2

B2

B1

A1

B

A

D1

C1

e or

w

P'

ε

ε

1

1

A2

C2

D2

B2

B1

A1

BA

D1

C1

C

qqA2

C2

D2

B2

B1A1

D1C1

B A M

CSL

(a) (b)

(c) (d)

Figure 2.5. Drained and undrained compression triaxial tests (schematic behaviour in CSSM).

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2) Lade & Duncan criterion:

(2.10)

where I1, I2 and I3 are the first, second and third stress invariants:

I1 � 1 � 2 � 3 (2.11a)

I2 � 1 2 � 2 3 � 3 1 (2.11b)

I3 � 1 2 3 (2.11c)

Aubertin & Simon (1996, 2000) also proposed a similar strength criterion for rocks.The criteria proposed by Matsuoka & Nakai (1974) and by Lade & Duncan (1975) are also shown in

Figure 2.6. It can be seen that the Matsuoka & Nakai’s criterion gives the same axisymmetric extension(b � ( 2 – 3)/( 1 – 3) � 1 in Figure 2.6) strength as Mohr-Coulomb whereas the Lade and Duncan’scriterion indicates a strength in axisymmetric extension that is slightly larger. Figure 2.7 shows resultsobtained on loose and dense Monterey No 0 sand. In both cases, the agreement with Lade & Duncan’smodel is very good.

From the compilation of numerous experimental results, Kulhawy & Mayne (1990) obtained the relative values of effective friction angles (as defined assuming the Mohr-Coulomb criterion) given inTable 2.1. It can be seen that the friction angle deduced from triaxial extension tests, ��te, is larger thanfriction angle deduced from triaxial compression tests, ��tc, typically by 10 to 25%. Lade (2002) indicatesthat Equation 2.10 gives a ��te/��tc ratio varying between 1.10 at low friction angles to 1.18 at high fric-tion angles. It thus seems that 1.15 can be considered as being a reasonable practical number for ��te/��tc.

3 LIMIT STATE CURVES OF NATURAL SOILS AND SOFT ROCKS (INFLUENCE OF ANISOTROPY)

3.1 Influence of anisotropy

The effective stress paths followed in undrained shear tests after consolidation in the normally consoli-dated range lie on the limit state surface. Thus, these stress paths are very close to the specific limit state

I

Icst1

3

3�

35

Figure 2.6. Failure criteria in octahedral (p� � cst) plane.

09031-02[1].qxd 18/Oct/02 12:09 AM Page 35

curves generated by the stress conditions applied during the consolidation stage. An idea of the influ-ence of consolidation stress conditions on the shape of the limit state surface can thus be obtained fromsuch tests. Ladd & Varallyay (1965) consolidated specimens of Boston Blue clay in the normally con-solidated range, under ratios of the radial effective stress �r to the axial effective stress �a equal toKonc � 0.54, 1.00 and 1/Konc � 1.85. The stress paths followed during the subsequent undrained com-pression and extension tests are shown in Figure 3.1. The following can be seen: (a) the shape of thelimit state surface is strongly influenced by the effective stress ratio applied in the normally consoli-dated range; (b) at large strains, the stress paths go to the same strength envelope; (c) for the same con-solidation conditions, the stress conditions at large strain are approximately at the same value of meaneffective stress p� in compression and in extension, indicating that the critical state concept is notaffected by initial anisotropy; (d) as the M parameter is not the same in compression and in extension,the deviatoric stress obtained at large strain in extension is smaller than that obtained in compres-sion. According to Equations 2.5 and 2.7 and the values of Mc and Me, the ratio should be equal to (3 � sin ��)/(3 � sin ��).

3.2 Limit state curves of natural clays

For natural soils that may be microstructured, i.e. with bonds at the interparticle contacts, determinationof the limit state curve has to be obtained from tests with initial conditions inside the limit state curve

36

Figure 2.7. Failure surfaces for loose and dense Monterey No 0 sand in octahedral plane (after Lade & Duncan,1975).

Table 2.1. Relative values of effective stress friction angles (after Kulhawy &Mayne, 1990).

Friction angle

Test type Cohesionless soil Normally consol. clay

Triaxial compression 1.00 ��tc 1.00 ��tcTriaxial extension 1.12 ��tc 1.22 ��tcPlane strain compression 1.12 ��tc 1.10 ��tcPlane strain extension 1.25 ��tc 1.34 ��tc

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(essentially in the elastic domain) and stress paths going towards and possibly outside the limit statecurve, where large plastic strains start developing. One possibility is to consolidate the soil close to insitu effective stresses and follow radial stress paths; another possibility is to combine isotropically oranisotropically consolidated drained or undrained triaxial (CID, CAD, CIU or CAU) tests andanisotropic compression (K � �r/ �a � cst) triaxial tests.

The clay chosen to show the determination of the limit state curve of natural clay comes from Saint-Alban (Tavenas & Leroueil, 1977). Undrained triaxial compression (CIU) and anisotropic consolidationtests were performed. Figure 3.2a shows the variation in deviatoric stress and pore pressure as a func-tion of strain for three CIU tests performed in the overconsolidated range. At the start of the test, theclay shows a strong resistance to compression until a peak is reached at a strain below 1%; beyond thepeak, large strains develop, associated with strain softening. The stress state corresponding to the pas-sage through the peak can be considered as a gross yield point (see Section 4) for the clay. The stresspaths associated with these three tests are shown in Figure 3.2c. The volumetric strains seen duringK � cst tests are shown as a function of ( �a � �r)/2 in Figure 3.2b. The passage from the overconsol-idated state to the normally consolidated state, that is the gross yield point, is well defined in all cases.The stress states corresponding to gross yielding are assembled in Figure 3.2c, so that the limit statecurve of the Saint-Alban clay from a depth of 3 m can be completely defined.

A number of remarks can be made in relation to the limit state curve and the behaviour of naturalclays. They are summarized in the following paragraphs.

The limit state curve is approximately centred on the Konc line in the ( �a – �r)/2 vs ( �a � �r)/2 dia-gram. This is certainly attributable to the anisotropy of stresses which exists during the formation of thedeposit (see Section 16.7, Figure 16.6 and Equations 16.9 to 16.11).

37

-300

-200

-100

0

100

200

300

0 100 200 300 400

σ' = σ 'ar

p' (kPa)

q (k

Pa)

σ' = 1.85 σ'a

r

Extension

Com

pres

sion

σ' = 0.54 σ'a

r

Figure 3.1. Stress paths followed during compression and extension undrained triaxial tests after consolidation inthe normally consolidated range under effective stress ratio �r/ �a of 0.54, 1.0 and 1.85 (after Ladd & Varallyay, 1965).

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The oedometer test also gives a gross yield point characterized by a vertical effective stress equal tothe preconsolidation pressure �p

2. Unfortunately, in these tests, the lateral stress is not usually mea-sured and the corresponding stress state is not known. The only thing that can be stated is that the stressstate lies on the line �a � �p. However, the definition of the preconsolidation pressure and experienceshow that the state of stress at �p is generally between points P and A in Figure 3.2c.

38

Figure 3.2. Triaxial testing on Saint-Alban (3 m) clay: (a) CIU tests; (b) anisotropic consolidation; (c) limit statecurve (after Tavenas & Leroueil, 1977).

2 In its geological meaning, the preconsolidation pressure is unique and constant. On the other hand, the “grossyield stress” separating small to moderate strains from large strains depends on strain rate, temperature andmicrostructure. It must thus be considered as a rheologic parameter. It could have a particular appellation suchas “vertical yield stress” or “apparent preconsolidation pressure”, but engineers often refer to it as “precon-solidation pressure”. So, this term is also used in this paper.

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In general, whatever the degree of microstructuration, the stress paths followed during undrained tri-axial tests (CIU or CAU) are at about constant p� (Skempton’s parameter A � �u/�( 1 � 3) � 0.33)when the OCR is about 2 to 3. The stress path tends to bend towards lower effective stresses (parameterA increases) when the overconsolidation ratio is smaller; it tends to bend towards larger effectivestresses (parameter A decreases) when the overconsolidation ratio is larger. These tendencies becomemore marked as the stress level increases and failure approaches. This latter aspect of soil behaviour,which has also been observed on reconstituted clays (Wroth & Loudon, 1967) was pointed out byCotecchia (2002*)3 and Leroueil et al. (2002*).

The upper part of the limit state curve is also the strength envelope of the overconsolidated soil. Itcorresponds to the Hvorslev surface described in Section 2 (Figures 2.2 and 2.3) for non-microstructuredsoils; it can be higher than the Hvorslev surface in microstructured soils (see Section 11). The undrainedshear strength of the overconsolidated soil is also on the peak strength envelope, at a location thatdepends on the overconsolidation ratio. Ladd et al. (1977) proposed to express the undrained shearstrength of the soil overconsolidated, su, as follows:

(3.1)

Where �ac is the axial consolidation stress, (su/ �ac)nc is the strength ratio of the normally consolidatedsoil and m a parameter that depends on the soil and the type of test. Parameter m is often close to 1.0,which implies that su/ �p is constant in the overconsolidated range. This characteristic is often used inpractice.

A general observation made is that the compression index obtained in anisotropic consolidation testsdepends on the effective stress ratio K. As indicated in Figure 3.2b, it is much larger for K values closeto Konc (equal to about 0.5) than for isotropic loading.

The limit state curves obtained at different depths at the same site have the same shape (Figure 3.3)and can be normalized using the preconsolidation pressure. In fact, all the behaviour inside the limitstate curve can be normalized using the preconsolidation pressure, at least for clays having the samegeological history and similar microstructure. Thus, samples taken from different depths and consoli-dated under stresses �c such that �c/ �p is the same, show the same relations ( �a – �r)/ �p � f(�) andu/ �p � g(�) when they are sheared (Figure 3.4). This type of normalization was included in the NSPconcept of Ladd (1969) and Ladd & Foott (1974), and is often used in practice (Ladd et al., 1977).

s sOCRu

ac

u

ac nc

m

′ ′

⋅

�

39

0

50

100

150

200

250

0 50 100 150 200 250 300 350p' (kPa)

q (k

Pa)

191 241 310 380

σ' (kPa)p

✛

✛

✛

✛✛

✛

Figure 3.3. Limit state curves for Winnipeg clay (from Graham et al., 1983b).

3 2002* refers to a contribution to this workshop.

09031-02[1].qxd 18/Oct/02 12:09 AM Page 39

Compiling limit state curves obtained on natural clays from 10 different countries, Diaz-Rodriguezet al. (1992) found the results summarized in Figure 3.5. Figure 3.5a shows extreme cases obtained onWinnipeg clay (��nc � 17.5°) and Mexico City clay (��nc � 43°). As shown in Figure 3.5b, the geomet-rical characteristics of the limit state curves ( �a – �r)max/2 �p and �piso/ �p (where �piso is the gross

40

Figure 3.4. Normalized stress-strain and pore pressure-strain relations in triaxial tests on intact Champlain sea clay(from Leroueil et al., 1985c).

Figure 3.5. Summary on the limit state curves of natural soft clays (after Diaz-Rodriguez et al., 1992).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 40

yield stress on the isotropic axis) respectively increases and decreases when �nc increases. This indi-cates that the shape of limit state curve of natural clays is essentially controlled by the friction angle ofthe normally consolidated soil or critical state friction angle. Characteristics of limit state curves of spe-cial interest for this Workshop have been added to Figure 3.5. It can be seen that they are in generalagreement with the data previously compiled. As indicated in Section 3.1 (Figure 3.1) and in Figure 3.6,the shape of the limit state curve is reflected by the stress paths followed in undrained compression andextension triaxial tests. In this latter figure from Ladd et al. (1977), the influence of plasticity and, indi-rectly, of the normally consolidated friction angle on the shape of the limit state curve was clearly indicated.

The failure criterion (Figures 2.6, 2.7, and 3.12) implies a strength in extension lower that thestrength in compression. This fact becomes even more important with anisotropy. As a result, the ratioof the undrained shear strengths obtained in extension and in compression increases from about 0.3 forlow plasticity soils to almost 0.8 for high plasticity clays as shown in Figure 3.7. As indicated inSections 16.5 and 16.6 (Figures 16.3 and 16.4), plasticity index is not a good parameter for describing

41

Figure 3.6. Effective stress paths followed in undrained compression and extension tests on normally consolidatedclays (after Ladd et al., 1977).

Figure 3.7. Undrained strength ratios in extension and compression versus plasticity index (modified afterKulhawy & Mayne, 1990).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 41

mechanical behaviour of clayey soils, and it can be expected that the scatter observed in Figure 3.7would be smaller if the strength ratios were to be plotted as a function of �nc.

The fact that the shape of limit state curves depends on effective stress history implies that it evolvesdepending on the effective stress path followed. This is called kinematic hardening. For a natural clayloaded under one-dimensional compression (horizontal strain equal to zero) in the normally consoli-dated range, the shape of the limit state curve would remain essentially the same; on the other hand,under an isotropic loading in the normally consolidated range, the shape of the limit state curve wouldprogressively evolve from its initial shape (Figure 3.5) to a shape corresponding to an isotropic material(Figure 2.3).

McCinty et al. (2000) performed triaxial tests on the intact Bothkennar clay taken from a depth of10.95–11.73 m and obtained the limit state curve shown with solid circles in Figure 3.8. Eight otherspecimens of this clay were isotropically consolidated to an effective stress of 210 kPa, 2.6 times theisotropic gross yield stress of the intact soil. They were then unloaded, and reloaded under differentstress ratios in order to define the new limit state curve. The results are also shown in Figure 3.8 withtriangles. They show a limit state curve that rotated towards the isotropic axis. Other samples consoli-dated in the normally consolidated range under different stress ratios and then unloaded and reloadedshow yielding conditions that generally confirm the rotation of the limit state curve with consolidationstress ratio and thus kinematic hardening. Futai et al. (2001) and Almeida & Marques (2002*) observeda similar rotation of the limit state curve of Sarapui clay when isotropically consolidated.

3.3 Limit state curves of other soils and soft rocks

Limit state curves have been determined on a variety of residual soils: volcanic agglomerate by Uriel &Serrano (1973); residual soil from gneiss by Sandroni (1981); residual soil from basalt by Maccarini

42

Figure 3.8. Limit state curves of Bothkennar clay, intact and after isotropic loading to high stresses (after McCintyet al., 2000).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 42

(1987); etc. An example is shown in Figure 3.9. Machado & Vilar (2002*) also show a limit state curvefor a weathered colluvium. In all these cases, the limit state curves appear to be fairly isotropic.

Limit state curves were also determined on soft rocks such as chalk and calcareous rocks (Addis &Jones, 1990; Elliot & Brown, 1985; Lagioia & Nova, 1995), mudstones (Ohtsuki et al., 1981), volcanictuff (Pellegrino, 1970; Aversa et al., 1991) and marls (Kavvadas et al., 1993, 2002*).

An interesting case was provided by Ohtsuki et al. (1981). These authors studied a weak volcanicmudstone with a high void ratio of 1.40 and a high apparent preconsolidation pressure of 5 MPa. Thelimit state curve, shown in Figure 3.10, is strongly anisotropic and very similar to those obtained on softclays (see Figure 3.2c). Leroueil & Vaughan (1990) hypothesized that the high void ratio of this mud-stone indicates that it has become microstructured early in its geological history while stresses were lowand water content high, and that the shape of the limit state curve of this strongly microstructured mate-rial still reflects the anisotropy to be expected of a freshly sedimented clay.

Another interesting case is provided by Lagioia & Nova (1995). These authors report triaxial testsperformed on a calcarenite from the Bari area, in Italy. The results are very peculiar, showing a markedtransition from rock-like to soil-like behaviour and are worth being described here. Figure 3.11a showsthe isotropic compression curve. It can be seen that the behaviour appears to be essentially elastic untilcollapse occurs at a mean effective stress of 2.4 MPa. After a significant decrease in void ratio at aboutthe same stress, the material regains stiffness. CID tests were also carried out. Figures 3.11b and c showtwo typical deviatoric stress-axial strain and volumetric strain-axial strain curves for confining stressesof 200 kPa and 1300 kPa. The corresponding stress paths are shown with dashed lines in Figure 3.11d.For the tests with the confining stress at 200 kPa, a peak is reached at an axial strain of about 1.5%

43

σ' -

σ'

(

)/

2(k

Pa)

ar

( )/2 (kPa)a rσ' + σ'

150

100

50

00 50 100 150 200 250 300

Critica

l stat

e

line

Figure 3.9. Limit state curve obtained for a residual soil from gneiss (from Sandroni, 1981).

Figure 3.10. Failure and yield of a porous silty mudstone (after Ohtsuki et al., 1981).

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(Point (a) in Figure 3.11b) and is followed by some strain softening. For the test with a confining stressof 1300 kPa, the deviatoric stress increases up to point (c) where there is yielding and breaking down ofbonds until point (d) under an essentially constant deviatoric stress. Then, at an axial strain of about 8%,q starts increasing until a maximum value is reached at large strain. The limit state curve in this case isclearly associated with the destructuration of the calcarenite. This limit state curve (Figure 3.11d) isessentially elliptic in shape and centred on the p� axis. Kavvadas et al. (2002*) also found a well-definedlimit state curve of elliptic shape for Corinth marl.

These observations and others summarized by Leroueil & Vaughan (1990), clearly show that the con-cept of yielding initially defined for reconstitued clays also applies to microstructured natural geoma-terials. The shape of the limit state curve of a given material reflects its geological history includingdiagenetic processes. Whereas it appears to be approximately isotropic for most residual soils and softrocks, it is clearly anisotropic for natural soft clays.

3.4 Three-dimensional aspects of limit state surface

The behaviour of soils and rocks described in this paper has been obtained mainly with �2 � �3, eitherin oedometer or triaxial tests, and with the axial stress axis coincident with the vertical direction of thematerial in in situ conditions. Few studies have been performed on the influence of the intermediateprincipal effective stress, �2, expressed by b � ( �2 � �3)/( �1 � �3), and of stress axis rotation charac-terized by �, the direction of the major principal stress to the vertical. The effect of stress axis rotationis examined in Section 6 on anisotropy. Hereunder, only the effect of b is considered.

Callisto & Calabresi (1998) tested Pisa clay in a True Triaxial Apparatus (TTA). The natural clay wasfirst consolidated to in situ stress conditions ( �a � 113.5 kPa; �r � 75.5 kPa). The overconsolidationratio was about 1.6. After consolidation, the clay specimens were subjected to radial effective stresspaths in a constant p� plane (p�� 88 kPa). Conditions at failure are shown in Figure 3.12a with dots andthe bold line. It can be seen that the failure line of Pisa clay is more rounded than that given by the

44

Figure 3.11. Triaxial tests on calcarenite: (a) isotropic compression; (b) and (c) examples of CID tests; (d) limitstate curve (after Lagioia & Nova, 1995).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 44

Mohr-Coulomb criterion. It is similar to the one given by Lade & Duncan (1975) criterion (Equation2.10 and Figure 2.6). Kirkgard & Lade (1991) and Lade & Kirkgard (2000) performed TTA tests onundisturbed, but normally consolidated, San Francisco Bay Mud and obtained similar results.

Boudali (1995) performed a series of TTA tests on the highly microstructured Louiseville clay. Afterisotropic consolidation to �c � 0.29 �p, radial effective stress paths were followed in a constantp�� 0.29 �p plane (also see Leroueil et al., 2002*). Results are shown in the octahedral plane inFigure 3.12b. The failure surface is rounded and well outside the Mohr-Coulomb failure surface corre-sponding to the normally consolidated friction angle of Louiseville clay. This clearly shows the effect ofmicrostructure on the 3D-limit state curve. Similar results were obtained by Mitchell & Wong (1973)and Wong & Mitchell (1975) on Ottawa clay.

3.5 Flow rule

Wong & Mitchell (1975), Graham et al. (1983b), Félix et al. (1985) and many others examined thedevelopment of plastic strains in natural clays. As an example, Figure 3.13 shows the plastic strainincrement vectors obtained at gross yielding of the Winnipeg clay (Graham et al., 1983b). The plasticstrain increment vectors shown on this figure are not perfectly perpendicular to the limit state curve.Other researchers also found results slightly at variance with normality, but with variable tendencies.So, considering how uncertain is the determination of plastic strain components and limit state curve,normality appears to be an acceptable approximation.

3.6 A simple yielding model

Stating that “a soft soil cannot be submitted to stresses in any direction higher than the previous stressesin that direction without yielding”, Larsson (1977) proposed that limit state curves of natural clays beschematized by four segments: two corresponding to the strength envelopes in compression and inextension; and two corresponding to �a � �p and �r � Konc �p. Limit state curves of natural soft clayshave been described in Section 3.2. Their shape reflects the stress anisotropy prevailing during soil depo-sition. However, a good fit between limit state curves and the schematic limit state curves suggested byLarsson (1977) is obtained when the �r � Konc �p segment is changed to �r � K* �p, with K* beingslightly larger than Konc. The following expression was proposed by Leroueil & Barbosa (2000):

K* � 0.85 � sin(0.46��nc) (3.2)

The validity of this approach is confirmed by Figure 3.14 where the schematized limit state curvesbased on Equation 3.2 are shown with broken lines for Winnipeg and Mexico City clays. K* values

45

Figure 3.12. Failure envelopes determined from tests in the TTA: (a) Pisa clay (after Callisto & Calabresi, 1998);(b) Louiseville clay at p�/ p� � 0.29 (after Boudali, 1995).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 45

larger than Konc have probably to be associated with the fact that Ko increases during secondary con-solidation or when strain rate decreases (Section 9.7). Also, limit state curves of soils are certainly morerounded than those given by the model, but this latter represents a simple and useful approximation. Theparameter K*, as defined by Equation 3.2, is thought to represent the distribution of inter-particle orinter-aggregate contacts in a soil and is more generally defined as KAL by Leroueil & Barbosa (2000).The parameter KAL represents the stress ratio on the Anisotropy Line (AL) and reflects soil anisotropy;as KAL does not seem to be significantly modified by bonding (microstructure) or partial saturation(Section 14.7), this model [called GFY (for Given Fabric Yielding) by Leroueil & Barbosa, 2000] seemsto be applicable to microstructured and unsaturated soils.

The GFY model is well adapted for describing kinematic hardening. The two segments of the limitstate curve, in between the compression and extension strength envelopes, intersect on the AL line, at

46

p

0.6

0.7

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0

p'/σ'

pq/

σ'

Mohr-Coulomb

GFY

Figure 3.13. Normalized limit state curve and plastic strain increment directions for Winnipeg clay (from Grahamet al., 1983b).

0.4

0.2

00 0.2 0.4 0.6 0.8

( σ'

- σ

' )

/ 2 σ

'a

pr

Winnipeg ( = 17.5˚)φ'

1.0

'φ'nc = 17.5˚nc

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8

( σ' + σ ' ) / 2 σ 'a r p

( σ ' + σ ' ) / 2 σ 'a r p

( σ'

- σ'

) / 2

σ'

ar

p

1.0

Mexico City ( = 43˚)'nc

nc = 43˚

AL

AL

(b)

(a)

φ' φ'

Figure 3.14. Yielding and GFY curves of natural saturated clays (after Leroueil & Barbosa, 2000).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 46

a given deviatoric stress when the soil is anisotropic or at zero deviatoric stress when the soil isisotropic. For example, when applied to the isotropic weathered colluvium studied by Machado & Vilar(2002*), the GFY model gives the limit state curve shown in Figure 14.22.

The authors also tried to apply the GFY model to the results obtained on Bothkennar clay and presentedin Figure 3.8. The GFY model captures relatively well the behaviour of the intact soil, even if the grossyield points indicate a KAL value larger than that indicated by Equation 3.2. For the soil consolidatedisotropically in the normally consolidated range, the GFY model captures the limit state curve in the com-pression domain (q � 0) well, but overestimates yield stresses in the extension (q � 0) domain. This is pos-sibly because the soil kept some memory of its initial fabric, even after isotropic consolidation to 2.6 timesthe isotropic gross yield stress of the intact soil. It also possibly shows the limits of this simple model.

Figure 3.13 shows the plastic strain increment vectors at yielding for Winnipeg clay. The dashed lineis the limit state curve given by the GFY model. It can be seen that the plastic strain increment vectorsare essentially perpendicular to this latter limit state curve, indicating that normality could also be appli-cable to the GFY model.

3.7 Some simple applications

The concepts of limit and critical states can be used for understanding the behaviour of natural soils andsoft rocks under a variety of test conditions (Leroueil & Vaughan, 1990). The cases of isotropically con-solidated drained (CID) triaxial tests and one-dimensional compression tests are described hereunder.

Figure 3.15 shows CID test results obtained on a Champlain Sea clay and on an oolitic limestone.Tests labelled “1” are at low confining stresses. They show a well-defined peak where the limit state

47

Figure 3.15. CID tests (from Leroueil &Vaughan, 1990).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 47

curve coincides with shear failure; at larger deformations, there is strain-softening and the materialmoves towards its critical state. Tests labelled “3”, at high confining stresses, show stiff behaviour untilthe limit state curve is reached, followed by a deviatoric stress progressively increasing towards the crit-ical state line. In tests labelled “2”, the stress path reaches the limit state curve and the critical state lineconcurrently, and the deviatoric stress remains constant after yielding. It is worth noting that these two

48

Figure 3.16. One-dimensional compression tests (from Leroueil & Vaughan, 1990).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 48

materials of completely different origin, and with strengths differing by more than two orders of mag-nitude, qualitatively present the same behaviour.

Figure 3.16 shows the compression curve and the corresponding stress path obtained from one-dimensional (Ko) tests performed on four different materials. Within their limit state curves all thematerials show stiff behaviour, with stress paths consistent with “elastic” materials. Theoretically,� �3/� �1 � �/(1 � �) for isotropic elastic materials with Possson’s ratio �. When the stress pathreaches the limit state curve, there is yielding and compressibility increases markedly. The stress paththen migrates towards the Ko line corresponding to Konc � (1 � sin ��). This is a transition phase; thestress path remains on this Konc line thereafter. The four materials which show these similar patterns ofbehaviour are soft clay, artificially bonded soil, marl and chalk, with apparent preconsolidation pres-sures of 75, 170, 1250 and 40000 kPa, respectively. Again, the behaviour is congruent although thematerials are very different.

This latter behaviour is confirmed by a series of triaxial tests performed at Université Laval on Saint-Louis-de-Yamaska clay. On this clay, with a preconsolidation pressure of about 180 kPa, two triaxial Kotests were performed after isotropic consolidation to 10 and 100 kPa respectively. The stress paths areshown in Figure 3.17. The three phases are clearly shown with elastic response (I–a), transition (a–b) andplastic response (b–c). However, while the effective stress ratio �r/ �a increases during the transitionphase in test 1 ( �c � 10 kPa) before reaching the Konc line, it decreases in test 2 ( �c � 100 kPa).

4 BEHAVIOUR INSIDE THE LIMIT STATE CURVE

4.1 Generalities

In previous sections, it has been assumed that the behaviour inside the limit state curve was elastic. It isnot the case. With the development of devices allowing the accurate measurement of local strains (e.g.Jardine et al., 1984; Lo Presti et al., 2001), it has become possible to examine in more detail the behaviourof soils inside their limit state curve. Jardine et al. (1991), Jardine (1992) and Hight & Higgins (1994)schematized this behaviour as shown in Figure 4.1. It can be described as follows for a stress path suchas IABCD:

– There is the outer yield curve Y3 (Figure 4.1a) that is associated with a change in fabric and coinci-dent with the limit state curve. Soils experiencing stress paths that reach this curve undergo largeplastic strains (Figure 4.1c). Inside Y3, strains are small to moderate.

– Within the inner yield curve Y1 surrounding the current effective stress conditions (zone 1), thebehaviour is linear-elastic and characterized by small strain elastic properties, in particular the smallstrain shear modulus Go.

– When a stress path crosses Y1, but remains inside zone 2 (between A and B), the behaviour is non-linear elastic.

– Between Y2 and Y3 (between B and C in zone 3), soil develops plastic strains. As indicated inFigure 4.1c, the ratio of plastic to total strain progressively increases as the stress path approaches C

49

Figure 3.17. Ko triaxial tests on Saint-Louis-de-Yamaska clay.

09031-02[1].qxd 18/Oct/02 12:09 AM Page 49

on Y3 and stiffness decreases. Passage through Y2, point B, also corresponds to the strain at whichpore pressures start to build up in resonant column or cyclic triaxial tests. From data compiled by LoPresti (1989) the strain on Y2 increases with Ip, and also with OCR.

– When Y1 and Y2 are crossed by the effective stress path, they are dragged with the effective stressstate. However, an elastic behaviour is obtained only if there is a sharp change in stress path directionor if the soil can benefit from some aging.

As an example, Figure 4.2 shows the yield curves Y1, Y2 and Y3 around in situ stress conditions forBothkennar clay (Smith et al., 1992; Hight et al., 2002b*).

The yield curves Y1 and Y2 are reached at strains that are smaller than 0.1%, and even smaller than0.001% in cohesionless or low plasticity soils. The study of soil behaviour in these zones thus necessi-tates equipment with extremely high resolution. In particular, to avoid system compliance and beddingand seating errors, it is essential to measure strains locally over the central part of the specimen. Readersinterested to get information and references on this aspect can refer to Lo Presti et al. (2001).

Small strain shear modulus, Go, can also be deduced from shear wave propagation velocity in soils,by using the following equation:

Go � �Vs2 (4.1)

where � is the mass density and Vs is the shear wave velocity.The shear wave velocity can be measured in the laboratory with piezoceramic bender elements

(Dyvik & Madshus, 1985; Viggiani & Atkinson, 1995b) or miniature geophones in the case of large sizetriaxial specimens or calibration chamber specimens (Nishio & Tamaoki, 1988; Stokoe et al., 1991). In situ, shear wave velocity can be obtained from seismic tests. The most widely used invasive techniques

50

Figure 4.1. Scheme of multiple yield surfaces and soil response (modified after Jardine et al., 1991; Jardine, 1992;and Hight & Higgins, 1994).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 50

are Cross Hole (CH), Down Hole (DH; Stokoe & Wood, 1972) and seismic cone (SC; Campanella et al.,1986) tests.

The small strain shear modulus can also be directly measured in resonant column (RC), torsionalshear (TS) or hollow cylinder (HC) apparatuses.

4.2 Small strain behaviour (inside zone 1)

4.2.1 Compliance matrix for cross-anisotropic materialsInside zone 1, soil behaviour is linear-elastic and can thus be characterized by a linear relationshipbetween the effective stress tensor and the strain tensor. The compliance matrix linking effective stressesand strains is as shown in Figure 4.3 for cross-anisotropic materials having the axis of symmetry in thevertical direction. This matrix comprises 8 parameters that will be discussed in the next paragraphs.

4.2.1.1 Young’s moduliThe elastic Young’s modulus Ex is defined as d �x/d�x where �x and �x are respectively the axial stressand the axial strain in the direction x. Examples are shown for a variety of geomaterials in Figure 4.4.

51

Figure 4.2. Yield surfaces identified for Bothkennar clay under in situ stresses (after Smith et al., 1992; Hightet al., 2002*).

Figure 4.3. Compliance matrix for cross-anisotropic materials (Tatsuoka & Kohata, 1995).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 51

It can be seen that Ev typically varies from 105kPa for normally consolidated clays to almost 108 kPa forhard rocks. In any particular direction x, Ex is a unique function of the normal stress �x in the x direc-tion, independently of the normal stresses acting in the other orthogonal directions. This was suggestedby Hardin (1978) and has been confirmed since. It is well illustrated by the set of data obtained onNerima gravel and reported by Kohata et al. (1997), (Figure 4.5). Ev and Eh can thus be written:

(4.2)

(4.3)

where, �o is a reference effective stress and (Ev)o and (Eh)o are Young’s moduli at this reference stress.

E (E )h h oh

o

m

�

′′

E Ev v ov

o

m

�

( )

′′

52

Figure 4.4. Summary of the effects of strain rate on the very small strain Young’s modulus EV (Tatsuoka et al.,1997; Tatsuoka, 2000).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 52

Figure 4.6 shows normalized Young’s modulus as a function of confining stress for a variety of unce-mented granular materials. It can be seen that the different curves are essentially parallel, correspondingto an m value approximately equal to 0.5.

From Equations 4.2 and 4.3, the ratio Ev/Eh can be written as:

(4.4)E

E

E

Ev

h

v o

h o

v

h

m

�

( )

( )

′′

53

Figure 4.5. Relationship between: (a) vertical Young’s modulus and vertical stress; and (b) lateral Young’s modulusand lateral stress, at isotropic stress states and triaxial stress states (Kohata et al., 1997).

Figure 4.6. Summary of relationships between elastic Young’s modulus Eo and confining pressure for uncementedgranular materials (from Kohata et al., 1997).

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Equation 4.4 shows the influence of the stress ratio ( �v/ �h) and of the inherent anisotropy reflected by(Ev)o/(Eh)o, which is the ratio of the moduli when stress conditions are isotropic. Figure 4.7 shows someEv/Eh ratios versus �v/ �h relationships for granular soils. From these curves, it can be seen that, astested, Ticino sand would be isotropic whereas Chiba gravel would be highly anisotropic with(Ev)o/(Eh)o � 2.5. Other (Ev)o/(Eh)o values can be found in the literature, in particular for clayey mate-rials (Pennington et al., 1997; Jovicic & Coop, 1997; Rampello et al., 2002*).

It is worth noting that Equations 4.2 and 4.3 are valid only for geomaterials that are not significantlymicrostructured. For microstructured soils or soft rocks, Equations 4.2 and 4.3 have to be modified sothat Ev and Eh may have finite values under zero confining stress. The following forms are proposed:

(4.5)

(4.6)

with m is smaller than 0.5 (see Section 11.2.3, Figures 11.5 to 11.7, and also Tatsuoka et al., 2002*)

4.2.1.2 Additional conditions provided by the symmetry of the compliance matrix and the isotropic properties in the horizontal plane

As indicated by Tatsuoka et al. (1997; 2001), the compliance matrix is often considered symmetric,which implies:

�vh/Ev � �hv/Eh (4.7)

and

Ghh � Eh/2(1 � �hh) (4.8)

E E Eh hmo h oh

o

m

� �

( )

′′

E E Ev vmo v ov

o

m

� �

( )

′′

54

Figure 4.7. Stress-system-induced anisotropy for sands and gravels (from Kohata et al., 1997).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 54

As shown by Stokoe et al. (1991) and Bellotti et al. (1996) for sands, and by Lo Presti et al. (1999)and Pennington et al. (1997) for reconstituted clays:

Gvh � Ghv (4.9)

which is expected for a continuum medium.However, Butcher & Powell (1995) and Pennington et al. (1997) indicated that Equation 4.9 may not

be valid for very stiff or layered clays.With Equations 4.7, 4.8 and 4.9 (when valid), the number of independent variables of the compliance

matrix decreases to 5: Ev, Eh, �vh (or �hv), �hh and Gvh. Very often Gvh is referred to as Go or Gmax thesmall strain shear modulus.

4.2.1.3 Shear modulusHardin (1978) suggested that for clays, the small strain shear modulus, Go, depends on applied stresses,void ratio and overconsolidation ratio (OCR). It has however been shown that the effect of OCR is, to alarge extent, taken into account by the effect of void ratio and could be neglected. The empirical equa-tion describing the influence of the controlling factors on Go can then be written as follows:

Go � SF(e)( �v . �h)n Pa

(1�2n) (4.10)

where F(e) is a void ratio function that will be described later on, n is a parameter indicating the influ-ence of stress, Pa is the atmospheric pressure and S a dimensionless parameter characterizing the con-sidered soil.

Figure 4.8 shows the shear modulus, normalized with respect to a void ratio function F(e) � e�1.3 asa function of ( �v � �h) for a variety of reconstituted clays mainly tested in RCT. It can be seen that nis close to 0.25 for most of the clays, except for some of the Japanese clays for which it appears to beslightly larger. Values slightly larger than 0.25 have also been observed by Jamiolkowski & Lo Presti(2002*) on Messina sand and gravel (n � 0.27) and by Miura & Yagi (2002*) on coarse-grainedTouhoro volcanic soil (n � 0.34). It may be worth mentioning that the materials showing an n valuelarger than 0.25 seem to have very high friction angles (��nc of 28–38° for Japanese clays (Tanaka,

55

Figure 4.8. Summary of relationships between normalized Go and applied stresses for reconstituted clays (Hight,1998).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 55

2002); 35° for Messina sand and gravel; and larger than 40° for Touhoro volcanic soil), indicating thatthis latter parameter could influence the value of n.

Also, as for Equations 4.2 and 4.3 for Young’s moduli, the form of Equation 4.10 applies only forsoils that are not microstructured. The reader is referred to Section 11 to find information on the influ-ence of microstructure on Go.

Figure 4.9, modified after Lo Presti (1989), shows Go values as a function of e for a variety of soils.The slope which reflects the void ratio function F(e), does not seem to be significantly influenced by thetype of soil. Lo Presti (1989) deduced from this figure the function F(e) � 1/e1.3. Several other voidratio functions have been proposed: Hardin (1978) proposed 1/(0.3 � 0.7e2); some are in the formF(e) � 1/ex, with x varying between 1.1 and 1.55 (Jamiolkowski et al., 1995a; Shibuya & Tanaka,1996). Some are of a different form, such as the one proposed by Romo & Ovando (1994) for MexicoCity clay which is expressed as a function of Ip and Ir (�(wL � w)/Ip).

S may vary with the type of soil and also, due to the form of Equation 4.10, with the void ratio func-tion. If, as in the functions previously mentioned, F(e) takes a value of 1.0 for a void ratio equal to 1.0,then S would be the Go value (in atmospheres) at a void ratio of 1.0 under an isotropic confining stressof 1 atmosphere. As shown on Figure 4.9, and from other data from the literature, S values are gener-ally between 200 and 500 for clays that are not significantly microstructured and between 300 and 750for sands. Weston (1996) and Brignoli (1997) examined the stiffness of Ticino sand and gravel andfound an influence of the uniformity coefficient Cu � (d60/d10). According to the parameters providedby Jamiolkowski & Lo Presti (1998), S, as defined in Equation 4.10, would decrease from 612 to 372when Cu increases from 2 to 74, thus essentially in the previously mentioned range.

Hardin (1978) suggested that for clays, the small strain shear modulus, Go, depends on appliedstresses, void ratio and overconsolidation ratio (OCR). Viggiani & Atkinson (1995a) showed that Gocould be defined as follows for isotropically consolidated reconstituted clays:

(4.11)

where p�r is a reference pressure, p�p is the stress at the intersection of the swelling line with the isotropicnormal consolidation line and A, n and m are non-dimensional soil parameters. The ratio p�p/p� is thusthe overconsolidation ratio defined in terms of isotropoic stresses.

4.2.2 Influence of stress history and stress pathWhen soil follows a stress path such as (1) in Figure 4.10, the yield curves Y1 and Y2 are dragged withthe stress state. Shortly after stopping at point A, the shapes of Y1 and Y2 are as indicated by the dashed

G

pA

p

p

p

p0

r r

np

m

′′′

′′

�

56

Figure 4.9. Normalized maximum shear modulus versus void ratio (after Lo Presti, 1989; from Jamiolkowski et al.,1991).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 56

line (1). If the stress path followed to reach A is (2), the shape of Y1 and Y2 is as indicated by the dashedline (2), different from (1). The small strain behaviour of soils is thus kinematic (Atkinson et al., 1990;Jardine et al., 1991; Jardine, 1992). However, with decreasing strain rate (see Section 9.3) and withmicrostructuration, zones 1 and 2 grow in size and may be less dependent on previous stress history(Tatsuoka et al., 1997; Clayton and Heymann, 2001).

4.3 Non-linear stress-strain behaviour (zones 1 to 3)

When stress conditions in a soil element move out of zone 1 (Figure 4.1), the stress-strain behaviourbecomes non-linear and non-elastic. The non-linearity depends on several factors including the type ofsoil, previous stress and strain history, stress path and strain rate. Each practical problem and associatedstress paths have thus to be considered individually. However, there are several aspects of stress-strainbehaviour that seem to be general and will be described in the following paragraphs.

It is first essential to specify some parameters used to describe soil stiffness. As shown in Figure 4.11,several shear moduli can be defined. Monotonic loading enables to define at a point such as P a secant

57

Figure 4.10. Development of Y1 and Y2 zones by different recent stress-time histories (from Tatsuoka et al., 1997).

Figure 4.11. Definition of soil stiffness.

09031-02[1].qxd 18/Oct/02 12:09 AM Page 57

modulus Gs and a tangent modulus Gt. In case of cyclic loading, these moduli are relevant only to theloading part of the first cycle. For the subsequent cycles, the defined shear modulus is implicitely theequivalent shear modulus Geq. As a consequence, the relationship between shear modulus and shearstrain defined in monotonic (static) loading test may be different from the equivalent curve defined ina cyclic loading test. As indicated in Section 9.3 regarding the influence of strain rate, this phenomenoncould be increased by strain rate effects. This is schematically shown in Figure 4.12a. Figure 4.12bshows an example obtained on Ticino sand isotropically consolidated to 49 kPa and subjected to cyclicand monotonic tests. It can be seen that at strains smaller than about 10�3, in the elastic range, the shearmodulus is essentially the same for all tests. At larger strains, however, the curves associated with cyclicloading tests are slightly above those obtained in monotonic loading tests. However, several authorshave not observed significant differences between the two types of curves (Tatsuoka et al., 1997).

Figure 4.13 shows the variation of the undrained secant modulus Eu as a function of axial strain for3 British geomaterials, Chalk, London clay and Bothkennar clay. The initial modulus value varies from

58

Figure 4.12. Modulus reduction for monotonic and cyclic loadings.

Figure 4.13. Undrained modulus as a function of axial strain for three different British geomaterials (after Clayton& Heymann, 2001).

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one material to the other, but also the general shape of the curves and in particular the strain at whichthe modulus becomes non-linear.

Figure 4.14 shows normalized shear strain modulus G/Go versus shear strain � for two extreme geo-materials, namely the highly plastic Mexico City clay and the Messina sand and gravel. WhereasMexico City clay shows shear modulus decay at a shear strain of about 0.1%, such a decay starts at astrain that is less than 10�3 % for the sand and gravel.

Examining the cyclic test results obtained by different researchers on a variety of clayey soils,Vucetic & Dobry (1991) produced Figure 4.15 showing normalized shear modulus as a function of thecyclic shear strain for plasticity index varying from 0 to 200. The tendency indicated by Figure 4.15 hasgenerally been confirmed since, in particular by the results obtained on the extreme soils that are theMexico City clay (Ip � 147–238) and the Messina sand and gravel (Figure 4.14). However, as alreadyindicated by Vucetic & Dobry (1991) and confirmed many times since, “It is important to note that soilshaving a very sensitive structure, such as quick clays or cemented sands, may have G/Go curves quitedifferent from those in Figure 4.15, independently of their Ip”.

Considering a linear strain threshold �1 at G/Go ratio of 0.95, d’Onofrio et al. (1998) defined the correlation between �1 and plasticity index shown in Figure 4.16. �1 typically increases from 0.001%for cohesionless materials (Ip � 0) to about 0.03% for clayey soils with Ip � 100%. The data obtainedon Mexico City clay (Romo, 1995, and Diaz-Rodriguez, 2002*) as well as on Messina sand and gravelare consistent with the tendency given by Figure 4.16.

59

Figure 4.14. Normalized shear modulus versus shear strain, curves for two geomaterials.

Figure 4.15. Relation between G/Gmax and for normally and overconsolidated clays (modified after Vucetic &Dobry, 1991).

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The bulk modulus, K � �p�/��vol is also non-constant (Smith et al., 1992; Zdravkovic & Jardine,1997). Results of probing tests performed on Bothkennar clay are shown in Figure 4.17a where bulkmodulus normalized with respect to p� is plotted as a function of the volumetric strain. They confirmthat K varies with strain, and also indicate that K is stress path dependent. This latter aspect is alsoshown in Figure 4.17b. K/p� is seen to be greatest along paths where q is constant (� � 0 and 180°) andsmallest when a change in q dominates (� � 70° and 110°), (Smith et al., 1992). Other data on non-linearity of bulk modulus for Gault clay and London clay are reported respectively by Dasari & Bolton(1998) and by Hight et al. (2002a*).

4.4 Other factors influencing stress-strain behaviour inside the limit state curve

In addition to the previously mentioned factors (applied stresses, type of soil, void ratio, stress history andstress path), there are other factors such as strain rate and time, temperature, microstructure, matric suc-tion and discontinuities that may also influence the small strain behaviour of soils. The influence of thesefactors will be examined later on in Sections 9.3 for strain rate and time, 10.3 for temperature, 11.2.3 formicrostructure, 14.8 for matric suction and 12 for discontinuities. It may however be of interest to men-tion here that, due to secondary compression and development of microstructure, soil stiffness generallyincreases with time. This is often quantified by the following parameter (Afifi & Richart, 1973):

(4.12)

where G0(t � tp) and G0(t) are the small strain shear moduli at the end of primary consolidation (t � tp)and at time t respectively. Values of NG are given in Section 9.3 and Figure 9.16.

Considerations on soil stiffness in in situ conditions and on relationships between characteristicsdeduced from different tests are presented in Part II of this paper, more precisely in Section 17.5.1, andin the paper by Hight & Leroueil (2002*).

4.5 Stress-strain curves and stress paths inside the limit state curve

4.5.1 Representation of stress-strain curveResearchers have looked for mathematical representations of the pre-failure stress-strain curve. Anattractive and simple one is the hyperbolic function (Kondner, 1963; Duncan & Chang, 1970). Such afunction can be written in the form:

( 1 � 3) � �/(a � b�) (4.13)

NG (t) G t t

G t t t/tG0 p

0 p p�

� �

�

0 1( )

( ) log ( )⋅

60

Figure 4.16. Correlation between plasticity index and threshold strain (defined at G/Go � 0.95) (after d’Onofrioet al., 1998).

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that can also be modified as:

�/( 1 � 3) � a � b� (4.14)

where a and b, parameters that can be easily defined in a diagram of �/( 1 � 3) vs � where Equations4.13 and 4.14 correspond to a straight line (Kondner, 1963). However, more accurate data show that inmost cases, the hyperbolic function cannot capture the soil response from very small to intermediatestrains, from zone 1 to Y3 (Tatsuoka & Shibuya, 1991; Fahey, 1998).

Another important practical aspect, already mentioned in Section 3.2 (Figure 3.4), is that, in homo-geneous deposits, for the same OCR, the stress-strain curves can be normalized with respect to the preconsolidation pressure.

61

Figure 4.17. Variation of K/p� during drained probing tests on Bothkennar clay: (a) Ktan/p� versus (�vo1; (b) varia-tion of tangent and secant K/p� at �vo1 � 0.01% and 0.1% (after Smith et al., 1992).

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4.5.2 Stress paths inside the limit state curveAn observation made on reconstituted clays (see Wroth & Loudon, 1967), but also on microstructuredclays (Cotecchia & Chandler, 2000; Cotecchia, 2002*; Leroueil et al., 2002*) is that stress paths corre-sponding to linear elastic behaviour, i.e. at essentially constant p�(Skempton’s A parameter � 0.33), areobtained in undrained tests in stress planes such that p� is equal to 0.3 to 0.5 �p, independently of themicrostructure of the soil.

5 BEHAVIOUR OF COHESIONLESS SOILS AND CRITICAL STATE

The state of sedimented clayey soil is primarily controlled by mineralogy, stress history and agingprocesses. The state of cohesionless soils depends not only on stress history, but also on the mode ofdeposition or of mechanical compaction, and is generally characterized by the relative density.

5.1 Behaviour of cohesionless soils under drained and undrained conditions

Because of their relatively high hydraulic conductivity, the drained behaviour of sands and gravelsattracted the most attention historically. Researchers became interested in the undrained behaviour ofsands and gravels when they realized that such hydraulic conductivity may not be sufficient to preventthe development of excess pore pressures and possibly failures (liquefaction under gravity or earthquakeloading).

5.1.1 Drained conditionsThe drained behaviour of sands received great attention in the sixties (Ladanyi, 1960; Rowe, 1962;Bishop & Green, 1965; Lee & Seed, 1967; Vesic & Clough, 1968). A very detailed study was performedon the Sacramento sand by Lee & Seed (1967). These authors performed 4 series of drained triaxialcompression (CID) tests at confining stresses varying from 0.1 to 13.7 MPa on sand specimens pre-pared at four different relative densities from 38 to 100%. Figures 5.1a and b show the principal stressratio and the volumetric strain as a function of the axial strain for the soil prepared at a relative densityof 38%, and Figures 5.1c and d show the same curves for the soil prepared at a relative density of 100%.

For the loosest soil (Figures 5.1a and b), the stress strain curves generally do not show any peak andthe volumetric strain is generally in compression. This behaviour is mostly associated with loose mate-rials and is referred to as “loose behaviour”. The test results obtained on the dense sand (Figures 5.1cand d) show, for confining stress smaller than 2 MPa, stress-strain curves generally reaching a peak fol-lowed by strain softening, and a volumetric strain that is negative. It is also worth noting that the rate ofdilation (d�vol/d�1) is at its maximum when the peak is reached (see arrows). This type of behaviour isreferred to as “dense behaviour”.

Closer examination of Figure 5.1a shows that, under small stresses ( �3c � 0.5 MPa) the soil withDr � 38% shows a “dense behaviour” with �1/ �3 ratio at the peak larger than the same ratio at largestrain. Figure 5.1c shows that the dense soil (Dr � 100%) does not present any peak, and thus presentsa “loose behaviour” under large stresses ( �3c � 4 MPa). For all the tests, �1/ �3 at large strains is aboutthe same, at a value of 3.4.

The general drained behaviour is schematically shown in Figure 5.2. At a given density, the soil hasa curved peak strength envelope and a strength envelope at large strains (critical state line, CSL) char-acterized by the slope M in the q–p� diagram or ��cs (Figure 5.2a). For different densities, the peakstrength envelopes are as shown in Figure 5.2b, but the CSL is unique.

From the studies reported by Lee & Seed (1967) and others, it appears that the drained shear behav-iour of sands depends mainly on density and stress. As previously indicated, the peak strength of cohe-sionless soils is associated with the maximum rate of dilation and reflects the energy necessary for thesoil specimen to change its volume.

Rowe (1962) proposed the following equation to link stress conditions during shearing, with the rateof dilation, including at large strains when there is no volume change (critical state).

(5.1)

where ( �1/ �3)crit is the stress ratio at critical state, �vol is the volumetric strain and �1 is the major prin-cipal strain. (�d�vol/d�1) is the rate of dilation.

′′

′′

⋅

�

�

�

�1

3

1

3 11

crit

vold

d

62

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63

Figure 5.1. Typical drained triaxial test results on Sacramento River sand: (a) and (b) respectively principal stressratio versus axial strain and volumetric strain versus axial strain for loose sand; (c) and (d) respectively principalstress ratio versus axial strain and volumetric strain versus axial strain for dense sand (after Lee, 1965).

Figure 5.2. Variation of drained shear strength envelope for sand with: (a) confining pressure, and (b) density.

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When stress conditions at the peak are corrected for dilation (Equation 5.1), the peak strengthenvelopes shown in Figure 5.2 collapse to come closer to the CSL. Rowe (1962), Lee & Seed (1967)and others mention that some shearing energy also has to be attributed to particle rearrangement andcrushing.

It is important to mention that, although the influence of dilatancy on strength has been mostly studied on sands, Equation 5.1 is general and also applies to other geomaterials.

Bolton (1986) examined drained test results obtained on 17 sands with friction angle at critical state,��cs, varying from 30 to 37°. To define the combined effect of relative density and stress, he proposed arelative dilatancy index defined as follows:

IR � Dr(Q � ln p�) � R � Dr(10 � ln p�) � 1 (5.2)

where Dr is the relative density, p� is the mean effective stress (in kPa in Equation 5.2), and Q and R areempirical factors found equal to 10 (for quartz and feldspar minerals) and 1 respectively. The Q valuedepends on the grain crushing strength and would decrease to 8 for limestone, 7 for anthracite and 5.5for chalk (Bolton, 1986).

Bolton (1986) found the following correlations for IR values between 0 and 4.For plane strain conditions:

��max � ��crit � 5IR (°) (5.3a)

For triaxial strain conditions:

��max � ��crit � 3IR (°) (5.3b)

The friction angles considered in these equations are secant friction angles obtained by dropping atangent from the origin on to a single Mohr circle of effective stress (c�� 0).

For both test configurations,

(�d�vol/d�1)max � 0.3IR (5.4)

Figure 5.3 illustrates Equation 5.3b. Test results obtained for relative density of about 0.5 and about0.8 are shown in the same figure. It can be seen that the agreement is quite good, with a typical depar-ture of less than 2°.

5.1.2 Undrained conditionsBy definition, there is no dilatancy in saturated soils when tested under undrained conditions. The rangeof possible soil response under such conditions has been summarized by Robertson & Ahmadi (2002*)and in Figure 5.4a, showing idealized stress-strain curves deduced from undrained compression tests onspecimens initially consolidated under anisotropic conditions. Line A represents the weakest type ofresponse in which the sample reaches a peak strength and then strain-softens to a minimum strength at

64

Figure 5.3. Dilatancy component for sands failing at various mean effective stresses (after Bolton, 1986).

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ultimate state. Line D represents the strongest type of response in which the sample continues to strain-harden throughout the test, eventually reaching a much larger strength at ultimate state. Lines B and Crepresent the two intermediate responses. Line B illustrates the type of response associated with limitedstrain-softening to a quasi-steady-state (QSS) and then strain-hardening towards ultimate state. Thechange in behaviour occurs on what Ishihara et al. (1975) call the “phase-transformation line”. Alongline C, the sample continuously strain-hardens, even if, once the initial peak is reached, the stress-straincurve shows a plateau. Types A and B behaviour present a minimum strength that is smaller than thepeak strength. Stress paths are shown in Figure 5.4b for tests A and D. After reaching the peak at pointP, below the critical state line, the stress path moves towards the ultimate state situated on the criticalstate line. As for test D, after showing a decrease in p�, the stress path moves up along the critical stateline towards its ultimate state. Soil response depends on void ratio, soil fabric, initial stress conditionsand the type of test (compression, extension or simple shear).

5.1.3 Strain path testsChu & Lo (1994) performed triaxial strain path tests with constant (d�vol/d�1) and stress path tests withconstant effective stress ratio, on saturated Sydney sand. Figure 5.5 shows some results in a q–p�

65

Figure 5.4. Possible responses to triaxial undrained monotonic loading of sandy soils (Figure a is from Robertson& Ahmadi, 2002*).

Figure 5.5. Strain path testing results: (a) Stress paths and d�vol /d�a � cst lines; (b) Relationship between effectivestress ratio and strain increment ratio, Equation 5.5 (after Chu & Lo, 1994).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 65

diagram. The effective stress path resulting from shearing along a constant strain increment ratio pathtest asymptotically approaches a constant stress ratio line. The corresponding slope in the q–p� dia-gram was termed (q/p�)asy. Chu & Lo (1994) showed that (q/p�)asy depends on the strain incrementimposed to the soil (Figure 5.5b). The relationship shown in Figure 5.5b corresponds to the followingequation:

(5.5)

These strain path test results apparently link in a more general manner than Rowe’s approach the rateof dilation to strength envelope (or mobilized stress ratio �1/ �3 or q/p�). However, Equation 5.5 foundby Chu & Lo (1994) is different from Equation 5.1 from Rowe (1962), which could indicate that theinfluence of rate of dilation is stress path dependent.

5.2 Critical state, steady state and state parameter

According to Roscoe et al. (1958), critical state could be defined as that ultimate state of a sample atwhich any arbitrary further increment of shear distortion will not result in any change in void ratio oreffective stress. Poulos (1981) defined the “steady state of deformation” as follows: “The steady stateof deformation” for any mass of particles is that state in which the mass is continuously deforming atconstant volume, constant normal effective stress, and constant velocity. The steady state of deforma-tion is achieved only after all particle orientation has reached a statistically steady state condition andafter all particle breakage, if any, is complete, so that the shear stress needed to continue deformationand the velocity of deformation remain constant.”

Although the two definitions appear very similar except, possibly, for the reference made to constantvelocity of deformation in the case of the steady state, there has been discussion on whether the criticalstate and steady state lines were the same or not (Casagrande, 1975; Poulos, 1981; Sladen et al., 1985;Alarcon-Guzman et al., 1988; Been et al., 1991). According to Been et al. (1991), “the differencesappear to lie in the methods of measurement. Critical state workers have generally relied on drained,strain-controlled tests on dilatant samples to determine the critical state. The steady state is measured inundrained tests, usually on loose (contractive) samples.” Been et al. (1991) performed drained andundrained monotonic triaxial compression tests on Erksak sand and found that the critical and steadystate lines were coincident. The authors believe that both critical and steady states are identical. In ae–log p� diagram, the corresponding critical state line (CSL) or steady state line (SSL) is often char-acterized by its slope (Figure 5.6).

To characterize the behaviour of sands, Been & Jefferies (1985) proposed the state parameter �defined as the difference between the current void ratio eo and the void ratio on the steady state line essunder the same mean effective stress p� (Figure 5.6).

q

pM

d

dasy.

vol

′

⋅ ⋅

� ��

�0

11

1

3

66

ψ = e - eo ss

o

ss

e

e

A

1λ

Vo

idra

tioe

log p'

Steady stateline (SSL)

Figure 5.6. Definition of state parameter (after Been & Jefferies, 1985).

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The state parameter � neglects a cohesionless soil characteristic considered as extremely importanthistorically, i.e. (emax � emin) in which emax and emin are respectively the maximum and the minimumvoid ratios of the sand, as defined by usual standards. To take this aspect into account, Konrad (1988)suggested a normalized state parameter, �N, defined as follows:

�N � �/(emax � emin) (5.6)

In a manner similar to sedimented clays which are well characterized by their overconsolidation ratio,sands could have been characterized by p�cs/p�o, when p�o is the current mean effective stress and p�cs isthe mean effective stress on the critical or steady state line at the same void ratio. As noted by Hird &Hassona (1986), this ratio is related to the state parameter. It can be shown that:

p�cs/p�o � exp(�2.3�/) (5.7)

The normalized stress paths shown in Figure 5.7 indicate that this could be a reasonable approach.Test 45 shows a “normally consolidated behaviour” with a stress path that goes towards its critical stateCS by decreasing p�; on the other hand, tests 103 and 113 show an “overconsolidated behaviour”, withstrain hardening towards the same critical state.

Previous comments are based on the assumption that the critical/steady state line is unique in an e–log p�diagram. Several investigators have however questioned this uniqueness:

– Performing undrained compression and extension triaxial tests on loose Ottawa sand prepared bywater pluviation, Vaid et al. (1990) observed larger shear-induced pore pressures, and consequentlysmaller p� values and strength values at large strain in the case of extension tests. They concluded thatthe critical state line is stress path dependent. This has been confirmed by Yoshimine et al. (1999),and also by Shibuya & Hight (1987) and Hight et al. (1987) who performed undrained hollow cylin-der tests on isotropically consolidated specimens of loose Ham River sand with b � ( 2 � 3)/( 1 � 3) � 0.5 and principal stress direction � from 0° to 90°. Hight et al. (1987) showed that thelarge strain strength is strongly influenced by �, progressively decreasing when � varies from 0 to 90°.

– On the basis of undrained compression triaxial tests, Konrad (1990, 1993) and Doanh et al. (1999)indicated that the critical state line is not unique, but depends on the consolidation stress. As schema-tized in Figure 5.8, for specimens consolidated at the same void ratio and over a range of stressesleading to contractant behaviour, relatively small confining stresses (test B) give large strain shearstrength values which are smaller than those obtained for higher confining stresses (test A).According to Konrad (1990, 1993), rather than being a unique line, the critical states define a bandin the e–log p� diagram limited by upper and lower limits referred to as UF and LF lines in Figure 5.8.

5.3 Soil liquefaction and instability

Sladen et al. (1985) gave a definition of liquefaction in the context of field conditions. Adapted to lab-oratory testing, liquefaction could be defined as a phenomenon in which there is a reduction in strength

67

3

2

1

00 1 2 3 4 5

p'/p'

q/p

' cs

cs

350/2 (2% fines)350/10 (10% fines)

45

37

103113

CS

Figure 5.7. Normalized stress paths for undrained triaxial tests on Kogyuk sand (after Been & Jefferies, 1985).

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observed after failure reached by monotonic, cyclic or shock loading. Whereas up to the point of fail-ure loading may be undrained, drained or partially undrained, the post-failure stage is generallyundrained or only partly drained. The behaviour shown by the test A in Figure 5.4, with a peak at P andstrain softening is a liquefaction process.

Sladen et al. (1985) studied the fine uniform Nerlerk sand (with different percentages of fines) andthe coarser Leighton Buzzard Sand. The specimens were prepared loose (apparently by moist tamping),isotropically consolidated and subjected to undrained triaxial compression. The results obtained in thedifferent series were very similar. Figure 5.9 shows effective stress paths normalized with respect to themean effective stress at the critical state, p�cs, at the same void ratio for Leighton Buzzard sand. For con-solidation stress p�o equal to about twice p�cs, CIU tests show almost no strain softening and reach thecritical state (CS on the figure) corresponding to an M value of 1.2 (i.e. ��cs � 30°). For higher nor-malized consolidation stresses, e.g. p�o/p�cs � 6.9, the deviatoric stress increases, reaches a peak at pointP and then decreases towards the critical state CS.

All the peaks of the normalized stress paths fall on the same straight line (Figure 5.9). In a q–p�–e dia-gram, this line forms a surface called the “collapse surface” by Sladen et al. (1985). The collapse surface is shown in Figure 5.10 (Sasitharan et al., 1993). It can be seen that it comes as a complementto the conventional critical state models. Alarcon-Guzman et al. (1988) and Sasitharan et al. (1993) confirm the concept of collapse surface but their results indicate that, rather than being generated by straight lines as suggested by Sladen et al. (1985), the surface would be curved and defined by mono-tonic undrained stress paths. The collapse surface would then be a state boundary surface followed by soil elements that reach it and are moving towards the critical state line.

Sasitharan et al. (1993) performed constant shear drained (CSD) tests, i.e. triaxial tests at constantdeviatoric stress and decreasing mean effective stress, on loose Ottawa sand. Figure 5.11 shows the

68

1

2

3

1 B C Ap'/p'

LF

UF

M1

1M

Failureenvelope

Peak envelope

log p'

A

CB

cba

e

UF lineLF line

UF

q/p

' UF

P

F

(a)

(b)

Figure 5.8. Conceptual framework for strain-softening behaviour of loose sand samples (after Konrad, 1993).

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69

Figure 5.9. Series of monotonic undrained triaxial compression tests on loose Leighton Buzzard sand showing thecollapse surface and the critical state (modified from Sladen et al., 1985).

Figure 5.10. Typical collapse surface in q–p�–e space (from Sasitharan et al., 1993).

Figure 5.11. State boundary or collapse curve at e � 0.809 and CSD test (q � 100 kPa) on Ottawa sand (afterSasitharan et al., 1993).

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results of one of these tests. The initial void ratio was 0.804 under p� � 300 kPa and q � 100 kPa. Thespecimen collapsed when p� reached 140 kPa; the void ratio was then 0.809. It can be seen fromFigure 5.11 that the stress conditions at that time were very close to the collapse line corresponding tothis void ratio, confirming the possibility of soil collapse at a mobilized friction angle much smallerthan the friction angle at failure, ��cs for loose sands. According to Sladen et al. (1985), the collapse sur-face is “the limit of stability if drainage is impeded under load-controlled conditions”.

From Sladen et al. (1985)’s data (Figure 5.9), it turns out that, if stress conditions on the collapse sur-face are described by an “angle of mobilization ��mob” (e.g. corresponding to M6.9 for the test consoli-dated at 6.9 p�cs in Figure 5.9), this angle of mobilization would decrease from ��cs for p�o/p�cs � 2 to avalue corresponding to M6.9 for p�o/p�cs � 6.9. Data presented by Castro et al. (1982) and Been &Jefferies (1985) also indicate a decrease of ��mob as p�o/p�cs increases. On the other hand, Vaid & Chern(1985) and Lade (1993) came to the conclusion that ��mob would be a constant, that the peaks obtainedin CIU and CAU tests would be on a line, termed the instability line by Lade (1993), intersecting theorigin. However, instability can only occur above a cut-off value of ( �1 � �3) that increases with den-sity (Vaid & Chern, 1985; Lade, 1993; Vaid & Eliadorani, 1998).

The state parameter � defined in Figure 5.6 has been used to evaluate the behaviour of soils. Soil ele-ments with state conditions below the CSL (� � 0) will always have a dilatant behaviour in undrainedtests whereas soil elements with state conditions above the CSL (� � 0) will have a contractant behav-iour. If � is large enough, some strain softening during undrained loading may occur. The state param-eter has also been used for interpreting in situ tests and predicting debris flows (Lee et al., 1988).However, � has been defined from undrained tests often performed after isotropic consolidation and is thus associated with a given type of test. It may not apply to other types of test. This can be illustratedby results of CSD (Constant Shear Drained) tests, triaxial tests performed at constant deviatoric stressand decreasing mean effective stress.

Anderson & Sitar (1995) and Anderson & Riemer (1995) performed triaxial tests on a clayey collu-vial soil from Briones Park, near San Francisco. Anderson & Sitar (1995) anisotropically consolidatedspecimens above the steady state line in a e–log p� diagram and carried out undrained compression(CAU) tests. They observed contractant behaviour, with an ultimate deviatoric stress generally smallerthan the initial deviatoric stress. Anderson & Riemer (1995) consolidated the same soil to approxi-mately the same conditions before subjecting it to an essentially constant deviatoric stress and decreas-ing mean effective stress (Stress path such as IY in Figure 5.12a). As shown in Figure 5.13, dilatancywas observed rather than the expected contractancy. Santos et al. (1996) observed similar behaviour ona residual soil from Rio de Janeiro.

Leroueil (2001) explained this behaviour as follows (see schematic Figure 5.12). If initial stress con-ditions are at point I, in Figures 5.12a and b, and if the mean effective stress is decreased, there is firstsome volume change of the soil, generally swelling, and yielding is reached at a point such as Y. Thislatter point can be on the critical-state line (CSL), as shown in Figure 5.12a, or above in microstructuredor dense soils, or below, on the collapse surface, as observed in loose sands (Figure 5.9). If initial con-ditions are at point I1, well above the CSL, the soil at yielding, Y1, is above the CSL (Figure 5.12b); for

70

Figure 5.12. State paths followed in CSD tests (schematic). In (a), Y could be above, on or below the critical-stateline, depending on the soil considered.

09031-02[1].qxd 18/Oct/02 12:09 AM Page 70

undrained post-failure conditions, the ultimate strength, at C1 is smaller than the initially applied shearstress, and yielding is followed by an undrained collapse and flow of the soil mass. If initial conditionsare at point I3, below the CSL, yielding will appear at Y3 (Figure 5.12b). The ultimate state in undrainedconditions, C3, then corresponds to strength larger than the shear stress at yielding and is thus stabilis-ing. For the intermediate case with initial stress conditions, I2, above the CSL and yielding, Y2, below(Figure 5.12b), the soil would be dilatant with a tendency to swell towards the CSL. The colluvium soilstudied by Anderson & Sitar (1995) and Anderson & Riemer (1995), and the residual soil tested bySantos et al. (1996) seem to correspond to case 2, with contractant behaviour in undrained CAU testsand dilatant behaviour in drained CSD tests. In such tests, the potential for soil instability is thus notcontrolled by the state parameter �I2 associated with initial state conditions but by the equivalent param-eter at yielding, ��I2 (Figure 5.12b). To avoid confusion, Chu et al. (2002) call �� the “modified stateparameter”.

It can be added that, if initial stress conditions had been at point I�2, with a much larger state param-eter, �I�2, the soil behaviour in a CSD test would have been the same, namely dilatant. The state param-eter is thus not a parameter that can characterize soil behaviour in CSD tests. This could be the same forother tests or stress paths.

This remark made by Leroueil (2001) led Chu et al. (2001, 2002) to examine soil instability in a dif-ferent way. These authors examined instability of marine dredged sand in loose to dense state condi-tions, associating instability to the development of large plastic axial and volumetric strains. CIU andCSD triaxial tests were performed. Figure 5.14 shows some results in q–p� and e–p� diagrams. The critical state line and the instability lines corresponding to �� values of 0.060 and �0.186 are also shownas references.

The behaviour of loose sand can be illustrated by test DR7 (also test DR10). The specimen was firstlysheared to point A along a drained path. The deviatoric stress at point A was q � 150 kPa. The meaneffective stress was progressively decreased along an essentially constant q path from A to B to C. Therewere little axial and volumetric strain development until point B where both axial and volumetric strainsstarted to develop at a faster rate. B is on the instability line for this soil (�� � 0.060). With furtherreduction in p�, the stress path moved further towards the CSL that would correspond to the failure linefor this loose specimen. However, the test had to be stopped when the axial and volumetric strains wererespectively 8% and 2% at point C. Therefore, instability with development of large plastic strains canbe observed under fully drained conditions. Leong (2001) and Chu et al. (2002) term this type of insta-bility “conditional instability” in contrast with undrained instability termed “runaway instability”.However, if the pore pressure cannot dissipate freely, the specimen would not be able to support theimposed q anymore and “runaway instability” would occur. This is in particular the case if a specimenis brought under drained conditions above the instability line and has its drainage valve closed; as indi-cated by Chu et al. (2002) and Lade (1995), instability can develop due to increase in pore pressure bycreep.

71

Figure 5.13. State paths in CSD tests performed on colluvial soil from Briones Park (from Anderson & Riemer,1995).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 71

Test DR39 (also test DR40) was conducted on a dense specimen under q � 303 kPa from initialstress conditions below the CSL (point D). As shown in Figure 5.14a, p� was progressively decreasedfrom D to E to F to G. Axial and volumetric strains started to increase at point E (Figure 5.14b) in a dilat-ant manner, at �� close to �0.186. As shown in Figure 5.14a, instability (defined as the development oflarge plastic strains) is above the CSL. From E, the mean effective stress continued to decrease until thefailure line associated with its rate of dilation was reached, at point F. If during this process, dilationcould not be accommodated, such as in undrained conditions, negative pore pressure would tend todevelop, leading to an increase in effective stress and the specimen would remain stable.

From CIU and CSD tests, Chu et al. (2002) were able to define q/p� on the instability line, (q/p�)ins,as a function of �� (Figure 5.15). The instability curve defined that way crosses the �� � 0 line at a q/p�value of 1.35, corresponding to the critical state friction angle. It can be seen that the instability curveseems to approach asymptotically some limiting q/p� values when goes towards large positive or nega-tive values. From the data available, (q/p�)ins would be approximately constant and equal to 0.8 for ��values greater than 0.08. This seems to be in agreement with (and possibly explains) the results of Vaid& Chern (1985) and Lade (1993) that show an essentially constant mobilized friction angle at peak,��mob, above a cut-off value of ( �1 � �3).

It is worth noting that the difference between instability and failure also exists in natural clays forwhich ��( �1 � �3)max is lower than ��( �1 � �3)max (Bjerrum & Simons, 1960) and that the concept ofcollapse surface could be applicable to geomaterials other than sand.

72

Figure 5.14. General instability behaviour of loose and dense sand along a CSD path (after Chu et al., 2002).

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5.4 Influence of specimen preparation and soil fabric

The influence of specimen preparation on the cyclic stress-strain behaviour of sands has been wellestablished (e.g. Mulilis et al., 1977; Miura & Yagi, 2002*, for coarse-grained volcanic soils). It is, how-ever, only recently that similar effects have been well documented for static behaviour. Vaid et al. (1995)studied the undrained behaviour of loose fine uniform Syncrude sand on specimens reconstituted bymoist tamping (MT), air pluviation (AP) and water pluviation (WP). Figure 5.16 shows comparativeresults obtained in simple shear tests performed on specimens with the same void ratio. The moisttamped specimen is highly contractive; the air pluviated specimen is also strain softening but to a muchsmaller degree. In contrast, the water pluviated specimen behaves in a strain hardening (dilative) man-ner. At a shear strain of 15%, the shear stress mobilized on the WP specimen is 10 times larger than theshear stress mobilized on the MT specimen. Vaid & Sivathayalan (2000) observed similar behaviourwhen performing undrained triaxial compression and extension tests on Fraser River sand.

73

Figure 5.15. Instability curve from undrained and CSD tests on sand (after Chu et al., 2002).

Figure 5.16. Effect of specimen reconstitution method on undrained simple shear response (after Vaid et al., 1995).

09031-02[1].qxd 18/Oct/02 12:09 AM Page 73

Hoeg et al. (2000) compared the undrained behaviour of natural silt and a silty sand tailings in bothundisturbed and reconstituted conditions. The natural silt was from a 10,000-year-old fluvial depositand the mine tailings were deposited hydraulically 5 years prior to the investigation. About half of the26 undrained compression triaxial tests presented were performed on reconstituted specimens, gener-ally prepared by moist tamping. In all cases, the undisturbed specimens showed dilative and ductilebehaviour, whereas in all but a few cases, the reconstituted specimens showed contractive behaviour. Atypical comparison is shown in Figure 5.17 for silt. Even if the relative density was smaller (e � 0.81,Dr � 66%) than that of the reconstituted specimens (e � 0.73, Dr � 73%), the undisturbed specimenshowed a dilative behaviour with continuous strength increase, whereas the moist tamped specimenshowed a low peak and brittle behaviour. Hoeg et al. (2000) also showed that with the same grain sizedistribution and void ratio, reconstituted silty sand tailings specimens had Go values that were approxi-mately 25% lower than those of undisturbed specimens.

Differences in behaviour are associated with differences in fabric between specimens prepared bydifferent methods or between undisturbed and reconstituted soil specimens. In particular, it is thoughtthat moist tamping generates aggregates of particles with relatively large voids in between and this canbe viewed as a potentially collapsible fabric. This is similar to what happens when soils are compactedat degrees of saturation smaller than the one at the optimum (see Figures 14.6 and 19.2 as well as relatedcomments). Vaid & Sivathayalan (2000) consider that water pluviation more closely replicates the fab-ric of water deposited in situ sands. Indeed, for two sands, they observed very similar undrained soilresponse on undisturbed and WP specimens.

Mimura (2002*) reports for 4 Japanese sands comparative triaxial monotonic loading and cyclicloading tests performed on undisturbed specimens sampled by the freezing technique and on soil spec-imens reconstituted to the same void ratio by air pluviation. In all cases, the undisturbed soil shows ahigher resistance, in particular to liquefaction by cyclic loading. This can be due to aging, but can alsobe due to differences in fabric.

74

Figure 5.17. Comparison of typical stress-strain curves and pore pressure development for undisturbed, moisttamped and slurried specimens of natural silt (after Hoeg et al., 2000).

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All these observations show that the behaviour of cohesionless materials is extremely sensitive to fabric.

5.5 Influence of anisotropy and stress axis rotation

This aspect is examined in Section 6.

5.6 Influence of grain crushing

Compression of granular materials to high confining pressures and subsequent shearing induce graincrushing (Lee & Seed, 1967; Vesic & Clough, 1968; Colliat-Dangus et al., 1988). This is illustrated inFigure 5.18b showing grain size distributions of Chattahoochee River sand (Vesic & Clough, 1968)after isotropic compression to 2.1, 21 and 63 MPa (dashed lines) and after shearing under confining

75

Figure 5.18. Drained triaxial shear tests performed on loose and dense Chattahoochee River sand: (a) Isotropicnormally consolidated zone and critical state line; (b) Grain size distributions after compression and shear (afterVesic & Clough, 1968).

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stress of 21 and 63 MPa. This phenomenon influences the measured friction angle (Lee & Seed, 1967);it also influences the critical state line in the e–log p� diagram. The compression lines for the loose and dense Chattahoochee River sand as well as the critical state line deduced from the drained triaxialshear tests reported by Vesic & Clough (1968) are shown in Figure 5.18a. Even if the critical state lineis approximate, there are several noteworthy observations: (a) it shows a knee somewhere between 1 and2 MPa, and this is probably associated with crushing (Figure 5.18b); (b) the critical state line and thenormal compression line are curved under high stresses, which is logical considering the fact that thevoid ratio becomes very small and cannot take negative values; (c) the ratio of the mean effectivestresses on the normal compression line, p�e, and on the critical state line, p�cs, at the same void ratioseems approximately constant, which is consistent with CSSM.

The effect of crushing on the critical state line has been examined by Been et al. (1991) (Figure 5.19a)and Konrad (1997). As schematically shown in Figure 5.19b from Konrad (1997), the slope of the crit-ical state line abruptly changes at a mean stress p�c corresponding to the crushing stress of soil particles.This crushing stress typically varies between 500 and 2000 kPa, depending on the mineralogy, the angu-larity and the grain size distribution of the material. The critical state line can be characterized by formean stresses smaller than p�c and c for higher stresses.

The bilinearity of the critical state line has consequences for the meaning of the state parameter �.Again, one may suggest that the p�cs/p�o ratio should be considered in place of or in combination with � to describe the mechanical behaviour of cohesionless soils.

Materials particularly prone to crushing are coarse-grained volcanic soils and carbonate sands thatare described in this Workshop by Miura & Yagi (2002*) and Coop & Airey (2002*) respectively. Whenuncemented, these materials behave essentially as sands. Shimizu (1999) and Miura & Yagi (2002*)indicate however that volcanic soils show some particular features that can be summarized as follows:

• �� is generally high, between 40 and 50°.• The friction angle has a tendency to decrease as the confining stress and thus crushing increase. The

evolution of crushing with confining stress depends on particle hardness.• Due to crushing, the intermediate effective stress �2 developed in plane strain conditions is generally

smaller than in other sands.• Due to crushing, Go may increase with stress more rapidly than in other sands. Due to the fact that

particles may contain internal voids, the void ratio functions such as those described in Section4.2.1.3 may not apply in coarse volcanic soils (Miura & Yagi, 2002*).

6 INFLUENCE OF ANISOTROPY

6.1 Introduction

Civil engineering construction almost invariably involves changes to the stress state in the ground,resulting in movements and requiring stability under the new stress regime to be considered. We are

76

o

cs

c

CSL

e

eVo

idra

tioe

Log p'

1

λ1

λ

O

ec

mine

maxe

Log p'Log p' Log p'o csc

Cψ

Logp'p'ocs

b) Schematic behaviour(after Konrad, 1997)

0.80

0.75

0.70

0.65

0.60

0.5510 10 10 10

Vo

idra

tioe

p' (kPa)

UndrainedLoad-contr., compr.Strain-contr., compr.Strain-contr., ext.

Drained, compr.

ψ > 0 ψ < 0

2 3 4

a) Erksak 330/0.7 sand (after Been etal., 1991)

Figure 5.19. Influence of crushing on critical state lines.

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familiar with these stress changes in terms of the changes in deviator or shear stress that may take place;an example is presented in Figure 6.1a as contours of (� 1 � � 3)/2p associated with a pressure, p,applied to a circular footing on the surface of an isotropic elastic half space; � 1 and � 3 are thechanges in major and minor principal stresses, respectively. We are probably less familiar with both thechanges in the directions of these principal stress increments and the relative magnitude of the interme-diate principal stress increment, � 2. Figure 6.1b shows contours of the direction of � 1, expressed as an angle, � to the vertical, and Figure 6.1c shows contours of (� 2 � � 3)/(� 1 � � 3), or �b, bothfor the same case of the uniformly loaded circular footing. (For each parameter the distributions aresymmetrical about a vertical axis through the centre of the footing.)

The principal stress increments shown in Figure 6.1 are superimposed on the initial stresses in theground, and the combination determines both the magnitude and direction of 1, 2 and 3. Figure 6.2shows the directions of 1 predicted to occur at failure around the failure surface beneath a longembankment constructed on soft clay, in which the initial stresses involved K0 ( �ho/ �vo) of 0.49.Significant deviations of the 1 direction from the vertical are apparent. In a soil with pronounced fab-ric, such as a laminated clay, the response of the clay will obviously vary around the failure surface withchanging direction of the major principal stress. How significant are the variations in response to load-ing direction in the more homogeneous soils which are the subject of this review, i.e. how significant is

77

1.5 0.5 0.5 1.50

0.5

1.0

0.1

0.150.2 0.20.15

0.1

0.2

0.25

0.3

0.3

0.25Z/r0

r0

r/r0

(a) Contours of ( - )σ ∆

σ ∆

σ ∆1 3

1

2p

(b) Contours of α

α

(c) Contours of b∆

+ ve

Figure 6.1. Description of stress changes beneath vertically loaded circular footing on isotropic linear elastic foundation (� � 0.45).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 77

78

0o

4o

12o

18o24o50o

80o

90o

Figure 6.2. Orientation of major principal stress along failure surface beneath embankment.

Axial strain (%)0 2 4 6 8 10 12

Prin

cipa

l str

ess

ratio

, R

=σ '

1/σ '

3

0

2

4

6

Bedding planes

σ1

α δ

0.0 0.1 0.2 0.3 0.4 0.5

σ 1-σ 3

(kP

a)

0

50

100

150

200

250

(a) 0-12%

(b) 0-0.5%

Local axial strain, (%)0.000 0.001 0.002 0.003 0.004 0.005

σ 1-σ 3

(kP

a)

0

2

4

6

8

10

(c) 0-0.005%

σc'=80kPa

OCR=1e0.05=0.654 - 0.657

90°

0°

45°65°

0°

45°90°

65°

Figure 6.3. Plane strain compression tests on air pluviated Toyoura sand (Park,1993).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 78

the anisotropy of stiffness and strength in soil? This question is addressed in this Section. The impor-tance of anisotropy in engineering problems is considered by Hight and Leroueil (2002*).

We are limited in our ability to measure the anisotropy of strength and stiffness in soils, by control-ling both the direction of the major principal stress, 1,which we shall define in terms of its angle to thevertical, �, and the relative magnitude of the intermediate principal stress, 2, which we shall define interms of b (�( 2 � 3)/( 1 � 3)). We can run triaxial compression tests, in which � equals 0° and bequals 0, or triaxial extension tests in which � equals 90° and b equals 1. Comparison of data from thetwo types of test on the same soil does not allow the effects of changes in � and b to be separated.

Initially data will be presented on the anisotropy of strength and the effect on it of b for reconstitutedsoils, obtained from four different pieces of apparatus: the true triaxial apparatus (TTA); the plane straincompression apparatus with flexible boundaries (PSC), used to test tilted samples, i.e. samples in whichthe initial direction of the bedding relative to the fixed direction of 1 can be varied; the DirectionalShear Cell (DSC); and the Hollow Cylinder Apparatus (HCA). The way in which the observed patternof anisotropy is likely to be modified by the fabric and structure of natural soils is then discussed, beforeintroducing the limited data on anisotropy in natural soils. A presentation of data on the anisotropy ofstiffness at small and intermediate strains in reconstituted and natural soils is then introduced.

6.2 Anisotropy in reconstituted soils

6.2.1 Initial, induced and evolving anisotropy in sandsFigure 6.3 presents the results from drained PSC tests on isotropically consolidated tilted samples ofToyoura sand to illustrate how anisotropy evolves during shear. The data is presented in three separateplots, covering increasing ranges of axial strain. At small strains (Figure 6.3a), the response is almostisotropic, with little difference between samples sheared with their bedding planes at different angles tothe major principal stress directions. As strain levels increase (Figure 6.3b), the responses diverge andanisotropy is apparent. At large strains (Figure 6.3c), there is strong anisotropy in stiffness and strength.The evolution of anisotropy involves different facets of structure coming into play. At small strains, onecan imagine that particle (or aggregate) contacts play a significant part; at larger strains, changing par-ticle arrangement and void closure play a part. Thus, anisotropy at small strains is unlikely to reflectanisotropy at large strains.

Data from PSC tests of this type define the anisotropy of plane strain strength in pluviated sands(Figure 6.4). The level of anisotropy reduces as relative density reduces and disappears at large strains.Variations in peak friction angle are very significant in a wide range of dense sands (Figure 6.5). At thispoint, it is perhaps appropriate to present a summary of the effect of b on peak friction angle, as measured

79

φ'o

300 15 30 45 60 75 90

35

40

45

50

Dr = 81%

Dr = 53%

φ'cv

σc'=100 kPa

φ'max

φ'max

Major principal stress direction αº

Figure 6.4. Anisotropy of �� in Toyoura sand in plane strain (Park, 1993).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 79

80

Major principal stress direction, αo (horizontal bedding)0 3 0 60 90

φ 'm

ax

35

40

45

50ToyouraSilver Leighton BuzzardSilica (Dr=50%)MontereyTicinoHostunKarlsruhe

σ3'=80 kPa OCR=1 Dr=80-90%

Figure 6.5. Anisotropy of ��max in air pluviated sands in plane strain compression (Park & Tatsuoka, 1994).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

b

35

40

45

50

55

60

φ ' (º

)

Lade&Duncan (1973) - dense Monterey No.0 sand, Dr = 98%; CP = 60kPaLade&Duncan (1973) - loose Monterey No.0 sand, Dr = 27%; CP = 60kPaOchiai&Lade (1983)- dense Monterey No.0 sand, Dr = 93%; CP = 100kPaSymes (1983) - dense HRS, Dr = 85%; p' = 200 kPa

Lam&Tatsuoka (1988) - dense Toyura sand, Dr = 80%, CP = 100 kPa; = 0º

Lam&Tatsuoka (1988) - dense Toyura sand, Dr = 80%, CP = 100 kPa; = 30º

Lam&Tatsuoka (1988) - dense Toyura sand, Dr = 80%, CP = 100 kPa; = 60º

Lam&Tatsuoka (1988) - dense Toyura sand, Dr = 80%, CP = 100 kPa; = 90º

Figure 6.6. The effect of b on peak friction angle (from Zdravkovic, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 80

in TTA and HCA experiments on isotropically consolidated samples (Figure 6.6). Included are a set ofcurves for Toyoura sand from Lam and Tatsuoka (1988) in which b was varied in tests with different incli-nations of the bedding planes to the vertical. The data from Symes (1983) was from HCA tests on denseHam river sand with � at 45°. It is apparent from Figure 6.6 that the peak value of �� increases when bincreases from 0 to 0.5 and remains constant or decreases slightly at b of 0.9 to 1.0. The effect of b becomesless at larger values of �, i.e. when the bedding plane inclination changes from horizontal to vertical.

The DSC, introduced by Arthur et al. (1977, 1981), is a plane strain device in which the direction of the major principal stress is controlled during shear by applying both normal stress and shear stressto four faces of a cube (Figure 6.7). The DSC can be used in two configurations, as shown in Figure 6.8.In the first (Figure 6.8a), it can be used to shear samples in a plane parallel to the bedding planes.

81

σb Soil Sample

Acrylic Triangular

Prism

Embedded Shot

for Radiography

0 50mm

σb

σa

σa

Reinforcing

rubbed pulling

sheets

Rigid Backing Plate

Unreinforced

rubber strips

Pressure bag

Pressure bag

retaining vanes

σb σb

σa

σa

τa

τa

τb

τb

Figure 6.7. Application of normal and shear stresses in the UCL directional shear cell (Arthur et al., 1981).

Direction of deposition

Elevation

First loadingdirection

Subsequentloading direction

Plan

(a) To examine induced anisotropy

Elevation

(b) To examine initial anisotropy

Direction of deposition

��

��ψ

Figure 6.8. Directional shear testing.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 81

When sheared in this plane with different directions of the major principal stress, samples of looseLeighton Buzzard sand were found to be isotropic. A typical response in this plane is shown inFigure 6.9a as the curve labelled “First loading cycle”.

The fact that anisotropy can be induced has been demonstrated in these drained DSC tests by firstapplying a loading cycle in the horizontal plane with the principal stress direction fixed, then reloadingthe specimen with the principal stress at different angles, �, relative to the direction of the first loading(Figure 6.8a). The response was no longer isotropic but varied with the direction of reloading, as indi-cated in Figure 6.9a. The induced anisotropy resulted in the response being stiffer and stronger thanoriginally for some directions of loading, but softer and weaker for other directions.

In the second configuration for the DSC, shear and normal stresses can be applied to control the prin-cipal stress directions in the vertical plane, i.e. the plane containing the bedding planes (see Figure 6.8b),to examine the initial or inherent anisotropy. The results of such tests on the same Leighton Buzzardsand are shown in Figure 6.9b. Clearly, the sand is strongly anisotropic in the vertical plane; the fact that the sand is also isotropic in the horizontal plane is a demonstration that the sedimented sand iscross-anisotropic.

Comparing Figures 6.9a and b, it can be seen that, for some directions of loading, the response in the vertical plane is stiffer and stronger than the initial response in the horizontal plane, and softer andweaker for other directions. In fact, the pattern of initial anisotropy in the vertical plane is similar to thatinduced in the horizontal plane by the first loading cycle.

6.2.2 Undrained anisotropy in isotropically consolidated loose sandsIn the hollow cylinder apparatus, a sample for which is shown in Figure 6.10, control over the majorprincipal stress direction, �, and of b, is achieved by independently varying the inner and outer cell

82

Major principal strain, ε1 (%)

0 2 4 6 8 10 12 14

Str

ess

ratio

, R=

σ'1/

σ'3

1

2

3

4

5α = 0o

40o60o 70o 90o

Isotropic plane

Str

ess

ratio

, R=

σ'1/

σ'3

1

2

3

4

5 ψ = 0 o

20o

90o

50o

70o60o 80o

First loading cycle

(a) Induced

(b) Initial

Figure 6.9. Induced and initial anisotropy in loose Leighton Buzzard sand (Arthur et al., 1981).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 82

pressures, pi and p0, the axial load, W, and the torque, MT. The HCA is the only apparatus in which both� and b can be independently controlled.

Figure 6.11 compares the behaviour of water pluviated and isotropically consolidated Ham river sandsheared undrained with b � 0 and � held at values between 0° and 90° (Shibuya, 1985). Note, firstly, themajor differences in peak resistance mobilised at intermediate strains. This peak in resistance is reachedwell before the friction angle is fully mobilised in each test. Note, also, the difference in stiffness and howthe character of the material changes. At � � 0°, which corresponds to a triaxial compression test, thematerial shows its strongest response, it is ductile and there is a tendency to dilate. The tendency to dilateat large strains reduces as � increases, and at � of 90° no such tendency appears to exist.

83

=

P o P 1

σ z

σ

τ

r

σ θ

zθ

σ 1

σ 2

3

α

α

M T

W

σ

Figure 6.10. Idealised stress conditions in the wall of a hollow cylinder sample.

Octahedral shear strain (%)

0 4 8 12

(σ1'+σ2'+σ3')/3 (kPa)

0 50 100 150 200

(σ1-

σ 3)/

2 (

kPa)

0

50

100α = 0o

15o

30o

45o

60o90o

α = 0 o

15o

30o

45o

60o90o

Figure 6.11. Undrained HCA tests on water pluviated Ham river sand. Isotropic consolidation, b � 0 (Shibuya,1985).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 83

The results of a similar set of tests in a different HCA on water pluviated Fraser River sand are shownin Figure 6.12 and the pattern of behaviour is almost identical.

Figures 6.11 and 6.12 illustrate the effect of principal stress direction on the undrained behaviour ofsand for the condition of b � 0. The effect of b can be seen in Figure 6.13 which presents undrainedHCA tests on sedimented Ham river sand with b � 0.5 and 1.0. Putting the data together fromFigures 6.11 and 6.13, we can examine in Figure 6.14 the variation in strength ratio, tp/p�, with � and b,where tp is the peak resistance at intermediate strains. It is clear that, in these isotropically consolidatedloose sands, the effect of � is more significant than that of b, when considering undrained resistance.There is a factor of more than 2 in terms of the anisotropy of undrained strength in these materials.

Stacking the effective stress paths observed in a series of HCA tests along an � axis, latched on toconventional stress path axes, as shown in Figure 6.15, defines a surface which portrays the initialanisotropy of the material. For normally consolidated material, this is the local bounding surface. Thesurface has several uses, one of which is illustrated in Figure 6.16. By tracking the effective stress path

84

(σ1'+σ2'+σ3')/3 (kPa)

0 50 100 150 200

(σ1-

σ 3)/2

(kP

a)

0

50

100

150

ε1-ε3 (%)

0 5 10 15

15o α = 0o

30o

45o

60o

90o

15oα =0o

30o

45o

60o

90o

e=0.89-0.91

Figure 6.12. Undrained HCA tests on water pluviated Fraser River sand. Isotropic consolidation, b � 0(Uthayakumar & Vaid, 1998).

00

50

50 100 150 200

100

(σ

1-σ 3

)/2

(kP

a)

(σ1'+σ2'+σ3')/3 (kPa)0 5 10 15

M1M2M3M4M5M6

α=0°

α=45°

α=0°α=15°

α=30°α=45°α=60°α=75° α=60°

α=90°

α=15°

α=30°

0

50

100(σ

1-σ 3

)/2

(kP

a)

Octahedral shear strain (%)

Figure 6.13a. Undrained HCA tests on water pluviated Ham river sand. Isotropic consolidation, b � 0.5 (Shibuya,1985).

M11M14M15M16

M11M14M15M16

0

50

100

(σ1-

σ 3)/

2 (

kPa)

0 50 100 150 200

(σ1'+σ2'+σ3')/3 (kPa)

α=30°α=30°

α=45°α=45° α=60°α=60°α=90°α=90°

0

50

100

(σ1-

σ 3)/

2 (

kPa)

0 5 10 15

Octahedral shear strain (%)

Figure 6.13b. Undrained HCA tests on water pluviated Ham river sand. Isotropic consolidation, b � 1.0 (Shibuya,1985).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 84

85

Major principal stress direction, α0

0 20 40 60 80

t p / σ

' vc

0.0

0.2

0.4

0.6b=0.3Isotropicb=0 b=0.5 b=1

K0

Figure 6.14. Anisotropy of undrained stress ratio (intermediate strains) in Ham river sand.

(σ1 - σ3)/2(kPa)

(σ1 ’ + σ

2 ’ + σ3 ’)/3 (kPa)

1530

4560

7590

α (deg)

50100

150

200

0

50

100

0

50

100

Figure 6.15. Local bounding surface for isotropically consolidated water pluviated Ham river sand, b � 0.

1530

4560

75

90

α (deg)

50100

150

200

0

50

100

0

50

100

(σ1 - σ3)/2

(kPa)

(σ1’ + σ

2’ + σ3’)/3 (kPa)

PEAK LINE

Figure 6.16a. Undrained principal stress rotation at constant t leading to failure.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 85

in a problem, it is possible to predict the response of the material to rotation of principal stress direc-tions. The example shown in Figure 6.16a is for an element which is sheared initially at � � 0°. At acertain level of shear stress, the principal stresses are rotated and the shape of the surface dictates thatthe material will fail at a certain value of �, simply as a result of principal stress rotation. In Figure6.16b, the initial principal stress direction is at 60°; subsequent rotation of the principal stress directionto a lower � brings the stress path inside the surface, resulting in a stiff response.

6.2.3 Collapse surfaceA second use of such surfaces of initial anisotropy, shown in Figure 6.17a, is that they help to identifyregions where the response is brittle, i.e., where the surface descends after reaching a peak, and wherethere is no dilation at large strains and even a tendency to collapse. In water pluviated sands, the sam-ples tend to dilate at large strains when sheared with low values of �. At high values of �, the behaviouris brittle at large strains and the local bounding surface descends from a peak point.

The key features of a collapse surface are summarised in Figure 6.17b, which shows a constant �section through the surface; these features are:

• an undrained approach to the collapse surface results in a brittle response; this occurs well before thefull angle of shearing resistance, ��, is mobilised

86

(σ1 - σ3)/2(kPa)

(σ1 ’ + σ

2 ’ + σ3 ’)/3 (kPa)

1530

4560

7590

α (deg)

50100

150

200

0

50

100

0

50

100

PEAK LINE

Figure 6.16b. Principal stress rotation leading inside the local bounding surface.

(σ1 - σ3)/2(kPa)

(σ1 ’ + σ

2 ’ + σ3 ’)/3 (kPa)

1530

4560

7590

α (deg)

0

50

100

0

50

100

50100

150

200

‘Dilatant’ region

Collapse

surface

Figure 6.17a. Extent of collapse surface in relation to complete local bounding surface for water pluviated Hamriver sand.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 86

• an approach under load maintained conditions results in collapse and runaway failure whether theapproach is undrained or drained, unless pore pressures can dissipate immediately during the col-lapse process

• drained strain-controlled approaches to the collapse surface allow the effective stress path to crossthe surface, although this is marked by the onset of large strains.

The surface in a constant � section was referred to as the liquefaction line by Shibuya and Hight(1991); others have referred to it as the collapse surface (refer to Section 5.3, Figures 5.9 and 5.10). Thekey thing to note is that, in water pluviated sands, the collapse surface only exists for a limited range of� values. Its location can be modified by induced anisotropy.

6.2.4 Undrained anisotropy in K0 consolidated loose sandsFurther evidence of induced anisotropy is available from a comparison of the results of hollow cylindertests run on water pluviated Ham river sand which has been consolidated under K0 conditions, with 1vertical (Figure 6.18), with the data presented earlier on the isotropically consolidated material. A directcomparison can be made in Figure 6.19 in which the solid lines are the effective stress paths observedin K0-consolidated samples sheared with different values of �, in most cases after undrained unloadingfrom the consolidation stress. The broken lines are the effective stress paths from the tests on isotro-pically consolidated samples. It can be seen that K0 consolidation distorts the anisotropy locally to thedirection of 1 during consolidation. Effective stress paths at high values of � are coincident. Undrainedstrength ratios at intermediate strains after K0 consolidation have been added to Figure 6.14; thisemphasises how the increase in strength induced by K0-consolidation magnifies the anisotropy. Nowthe maximum difference in strength is a factor of 3.

6.2.5 Undrained anisotropy in K0 consolidated reconstituted siltZdravkovic and Jardine (2000) describe the anisotropy of undrained shear strength and yield of a denseK0 consolidated quartzitic silt (HPF4), formed by pluviation and investigated using an HCA. The silt isan industrially produced rock flour with highly angular and elongated particles. The results of the testson the normally consolidated silt, carried out with b � 0.5, are presented in Figure 6.20. Note that toexplore the shape of the three-dimensional bounding or limit state surface it was necessary to partiallyunload some of the samples before shearing with the principal stress direction fixed. The initialresponse of the silt was stiff and highly non-linear. Yield at the Y3 surface was reached with little changein p�, after which there was a stable contractant stage until the phase transformation point was reached,

87

Failure line

Collapse

surface

(-

)/2

σσ

13

Undrained

Drained

Drained

Load controlled -drained -undrained

Strain controlled

( + )/21 3σ’σ’

Figure 6.17b. Constant � section through collapse surface showing conditions leading to collapse.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 87

referred to as Y4 yield by Zdravkovic & Jardine (2000). After phase transformation all samples showeda strong tendency to dilate.

Figure 6.21 summarises the variations with the major principal stress direction, �, of the frictionangle, ��, and the normalised shear stress, t, at phase transformation. Also shown are the correspondingvalues from triaxial compression and extension tests. There is clearly a major effect of b on �� at phase

88

ε1-ε3 (%)

0 4 8 12

(σ1'+σ2'+σ3')/3 (kPa)0 50 100 150 200

(σ1-

σ 3)/

2 (

kPa)

0

50

100

150

α=0o

15o

45o

30o

90o

K o consolidation

α=0o

15o

45o

30o

90o

Figure 6.18. Undrained HCA tests on water pluviated Ham river sand. K0 consolidation, b � 0.3 (Shibuya, 1985).

(σ1'+σ2'+σ3')/3 (kPa)

0 50 100 150 200

( σ1-

σ 3)/2

(kP

a)

0

50

100

α=0o

15o

45o

90o

Ko consolidation (b=0.3)

Isotropic consolidation (b=0.5)

0o

15o

45o

90o

Figure 6.19. Comparison of anisotropy in isotropically and K0 consolidated Ham river sand.

0 50 100 150 2000

50

100

150M1M2M3M4M5M6

α=0o

α=15o

α=30o

α=45o

α=70o

α=90o

(σ1-σ

3)/2

(kP

a)

(σ1'+σ2'+σ3')/3 (kPa)

Figure 6.20. Undrained HCA tests on pluviated HPF4 silt. K0 consolidation, b � 0.5 (Zdravkovic & Jardine, 2000).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 88

89

0 15 30 45 60 75 90

α (º)

20

25

30

35

40

45

50

55

60

φ' (

º)

TXC

TXE

b = 0.3

b = 0.5; shearing after K0 consolidation

PT values of φ'

estimated plain strain envelope

b = 0.5; shearing after inclined consolidation

PT values of φ'

PT = Phase Transformation

Figure 6.21a. Anisotropy of �� mobilized at phase transformation in K0 and �c consolidated HPF4 silt.

0 15 30 45 60 75 90

α (º)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

t pt

/ pc'

TXC

TXE

also b = 0.3

b = 0.5; inclined consolidation to p0' = 400 kPa

b = 0.5; K0 consolidation to p0' = 200 kPa

Figure 6.21b. Anisotropy of normalized shear stress at phase transformation in K0 and �c consolidated HPF4 silt.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 89

transformation, being 52–53° at b � 0.5, 42° at b � 0.3 and 32° at b � 0 for � � 0°. There is also sig-nificant anisotropy in �� and t/p�0 at b � 0.5.

6.2.6 Undrained anisotropy in K0 consolidated reconstituted Boston Blue clayThe undrained anisotropy of reconstituted Boston Blue clay has been investigated in the DSC byO’Neill (1985) and Seah (1990), testing clays with OCRs of 4 and 1, respectively. The results ofO’Neill’s experiments are presented in Figure 6.22. These confirm that the sedimented clay is cross-anisotropic and that the pattern of anisotropy in the vertical plane is similar to that seen in the pluviatedsands. Again, strengths and stiffnesses are higher than in the isotropic plane for low values of �, andlower for high values of �. The source of anisotropy is differences in dilational tendencies in this over-consolidated clay, rather than any differences in ��.

90

Figure 6.22. Constant volume DSC tests on resedimented Boston Blue Clay. K0 consolidation, OCR � 4 (O’Neill,1985).

Major principal stress direction, α0

0 20 40 60 80

t p / σ

' vc

0.0

0.1

0.2

0.3

0.4

Ham river Sand

HK clayey sand

KSS clay-silt-sand mix

Boston Blue Clay (plane strain)

HPF4 silt

Figure 6.23a. Anisotropy of undrained peak strength ratio in K0 normally consolidated soils (Shibuya, 1985;Menkiti, 1995; Zdravkovic, 1996; O’Neill, 1985).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 90

6.2.7 Typical levels of anisotropy in reconstituted soilsFigure 6.23a summarises the data that has been obtained at Imperial College and MIT on anisotropy of undrained strength in K0-normally consolidated reconstituted sands, clays and silt. First of all, notethe similar pattern of anisotropy in each material, but how the variation is least in the artificial clay KSSand greatest in the water-pluviated granular materials. Undrained strengths can vary with � by a factorof up to 3 in these reconstituted soils.

Figure 6.23b compares the anisotropy in peak �� for dense Ham river sand (HRS), dense silt (HPF4),a clayey sand (HK), and the artificial clay (KSS). The clay is almost isotropic in terms of �� but thereis strong anisotropy in the granular materials, the level of which depends on their density.

A more complete description of the variation in normalised undrained strength, tp/ �vc, in Ham riversand is presented in Figure 6.24, using axes of b and �; a similar presentation is used in Figure 6.25 forthe variation in �� in dense Toyoura sand.

91

Major principal stress direction

0 20 40 60 80

φ 'o ul

t

30

40

50

HK (b=0.5) KSS (b=0.5)HRS (b=0.3) HPF4 (b=0.5)

Figure 6.23b. Anisotropy of ��ult in K0 normally consolidated soils (Shibuya, 1985; Menkiti, 1995; Zdravkovic,1996; O’Neill, 1985).

0

0.5

1

b

30

60

90

σ (deg)

0

0.25

0.5

0

0.25

0.5/σ’vc

TriaxialCompression

TriaxialExtension

t

Figure 6.24. Variation of tp/ �vc with b and � in Ham river sand, isotropic consolidation.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 91

6.2.8 Anisotropy induced by ac consolidationIn an elegant series of tests on the HPF4 silt, Zdravkovic (1996) has generalised the findings regardingthe anisotropy induced by non-isotropic consolidation. She has shown that consolidation along constantratio stress paths, but with different directions of the major principal stress (paths R0, R15, R30, R45,R70 and R90 in Figure 6.26), locally modifies the anisotropy of volume change behaviour, and, there-fore, of undrained response. Jardine et al. (1997) introduce the term �c to distinguish consolidationpaths with different principal stress directions during consolidation. In Figure 6.21b, the anisotropy ofundrained strength at phase transformation, tpt, normalised by the mean effective stress during consoli-dation, p�c, has been shown for the case of K0 consolidation to p�0 of 200 kPa, i.e. with �c equal to 0.These strengths may be compared with those measured at the same � values, but after consolidation top�c of 400 kPa with �c equal to the value of � during subsequent undrained shearing. In each case, sim-ilar levels of strength are induced in the �c direction, but at the expense of increased brittleness.

In the dense silt, the induced anisotropy is restricted to the behaviour at intermediate strains, i.e. inbehaviour up to phase transformation. The values of �� mobilised at large strains are relatively unaf-fected by the value of �c, (Figure 6.21a) but remain a function of the initial fabric and anisotropy. Thisconfirms that, in this dense granular material, fabric changes are relatively subtle and likely only toaffect response at small and intermediate strains, and then only local to the direction of 1 during con-solidation.

92

0

0.5

1

b

0

30

60

90

α (deg)

35

40

45

50

35

40

45

50φ’

TriaxialExtension

TriaxialCompression

Figure 6.25. Variation in peak friction angle with b and � in Toyoura sand, isotropic consolidation.

0255075

100125150

0100

200300

400

2040

6080

t (kP

a)

p' (kPa)Major principal stress direction, (o)

R0 R15 R30 R45 R70 R90

Figure 6.26. Consolidation stress paths for �c consolidation, R-series tests (Zdravkovic, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 92

6.3 Anisotropy in natural soils

6.3.1 Modifications to the pattern of anisotropy due to fabric and structureWhile the reconstituted materials that have been considered to date have their own particular fabric,related to the way in which they were reconstituted, they are not necessarily representative of the fabricsof natural sands and clays. Since anisotropy of stress-strain-strength behaviour is closely linked to fab-ric, it is necessary to consider the structure – that is the fabric, lithification and bonding – of naturalmaterials – and see how that affects the pattern of anisotropy observed in reconstituted soils. Hight(1998) speculated on the potential effects of depositional and post-depositional processes on theanisotropy of undrained strength in natural soil; taking the solid lines in Figure 6.27 as representing theanisotropy of the reconstituted soil, he suggested that they might be modified to the dotted and brokenlines. He presented evidence of increased anisotropy in soft clays as a result of ageing and diagenesis,which would distort the picture as shown in Figure 6.27a. In soft laminated or varved clays, it was sug-gested that more extreme levels of anisotropy were likely, in both drained and undrained strength, withminimum values occurring with � between 45° and 70°, as shown in Figure 6.27b. It was further sug-gested that burial compaction of clay-rich clays could lead to similar levels of fabric anisotropy and thusto similar levels of strength anisotropy. In silt-rich clays this particular anisotropic phenomenon was

93

Cu p

/ σ '

vc

0.0

0.1

0.2

0.3

0.4

ReconstitutedContact ageingIsotropic cementingAgeing

Reconstituted

Major principal stress direction, α0

Cu p

/ σ '

vc

0.020 40 60 800

0.1

0.2

0.3

0.4

Ageing

Bedding compactionlamination

Reconstituted

Figure 6.27. Potential effects on anisotropy of undrained peak strength ratio of ageing, bonding, bedding and compaction.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 93

less likely to occur. The presence of low strength shear surfaces or discontinuities would also modifyanisotropy. The cross-anisotropic properties of reconstituted sedimented soils would be modified byinclined bedding or by non-isotropic horizontal stresses, which would induce azimuthal anisotropy. Asshown in Figure 6.27a, the possibility of isotropic cementing was noted.

This speculation was aided by the existence of data from naturally bonded or cohesive materials inwhich anisotropy could be examined in tests on samples cut at different inclinations and loaded in com-pression. Figure 6.28 shows the results of two such investigations in mudrocks. Figure 6.28a presentsresults showing the anisotropy of uniaxial compressive strength in a slate, which, of course, presents anextreme example of fabric anisotropy and resulting strength anisotropy. The example in Figure 6.28bis of another mudrock, the Sagamihara mudstone, which shows only minor anisotropy of undrained tri-axial compressive strength; this mudrock has not experienced the same pressures as the slate and,

94

(b) Anisotropy of undrained triaxial compressive strength ofSagamihara mudstone (after Tatsuoka et al, 1997)

Inclination of bedding to vertical (degrees)0 45 90 135 180

Und

rain

ed tr

iaxi

al

com

pres

sive

str

engt

h (M

Pa)

0

2

4

6

8

10

(a) Anisotropy of uniaxial compressive strength in a slate (after Hoek, 1965)

Inclination of bedding to vertical (degrees)0 30 60 90

Uni

axia

l com

pres

sive

stre

ngth

(M

Pa)

0

50

100

150

σ1 σ1 σ1 σ1σ1

Figure 6.28. Anisotropy of strength in mudrocks.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 94

because of its significant silt content, has not developed a strongly anisotropic fabric. Two examples fornatural clays are presented in Figure 6.29; data for the heavily overconsolidated London Clay indicatesundrained strengths that are higher for horizontal loading, while data for the lightly overconsolidatedWelland Clay shows undrained strengths higher for vertical loading.

6.3.2 Investigation of anisotropy of a natural clay in an HCAInvestigations of the complete anisotropy of stiffness and strength of a natural soft clay are rare. Onereported by Leroueil et al. (2002*) on the Louiseville clay is cited here. A series of tests in a hollowcylinder apparatus of the type used by Saada & Ou (1973) were performed (Leblond, 2002). The spec-imens tested were taken from a depth of about 10 m. The specimens were carefully trimmed to an exter-nal diameter of 70 mm, an internal diameter of 50 mm and a height of 160 mm. The specimens weremounted in the HCA and consolidated under an isotropic stress equal to 0.3 �p. They were finallysheared in undrained conditions with the axis of the major principal stress at an angle � to the vertical

95

0 0 60 90

α (°)

0

0.5

1

1.5

Cu

0 / C

uv

London Clay (Ward et al, 1965)Welland Clay (Lo, 1965)

3

Figure 6.29. Anisotropy of strength in two natural clays.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 95

that varied from 0° (conventional compression) to 75°. Other tests were performed at � angles of 0°, 22°and 29° after isotropic consolidation to 0.19, 0.44 and 0.54 �p. As the inner and outer cell pressures were equal in the HCA used, � was linked to the intermediate principal stress 2 by the equation:( 2 � 3)/( 1 � 3) � sin2�.

Well-defined failure surfaces were observed in all tests, and the angle of these surfaces to the hori-zontal, �, was measured. The � values are plotted as a function of � in Figure 6.30. As can be seen, theshear surface was at an angle � of 56° for � � 0, decreasing to 22° at � � 29°, and then abruptlydropped to 0° for larger � values. According to the Mohr-Coulomb criterion, � is related to � as follows:

(4)

This criterion has been shown on Figure 6.30 for a friction angle of 28°. While the soil behaviourapproximately follows Equation 4 for � values smaller than 30°, there is a significant variance from theMohr-Coulomb criterion for larger � values.

The tests performed at � � 0, 22 and 29° after consolidation at different effective stresses gave thestrength envelopes shown in Figure 6.31. For the tests performed at � � 30°, stress conditions at fail-ure on the horizontal failure plane were calculated and are plotted on Figure 6.31. It can be seen that thisgives a strength envelope that is below the strength envelopes deduced from the � � 0–29° tests. Thisdemonstrates that Louiseville clay has lower strength characteristics on the horizontal plane than oninclined planes. It has been observed, using scanning electron microscopy, that the relief is more pro-nounced on the vertical plane than on the horizontal plane (Lapierre et al., 1990). The differencebetween the strength envelopes for horizontal and inclined planes may be due to this difference inroughness.

The Corinth Marl described by Kavvadas et al. (2002*) is reported to be isotropic because of the highstrength of isotropic bonds, the presence of non-platey silt size particles, and because of the early devel-opment of bonding, which has maintained an open structure and inhibited the development of stressinduced anisotropy.

� � ��

� �452

° ′

96

70

60

50

40

30

20

10

0

-10

-20

�

βσ'

Mohr - Coulombcriterion ( ('φ = 28̊θ, ˚

β ,˚10 20 30 40 50 60 70 80

1

Figure 6.30. Inclination of failure surface with horizontal, �, versus the angle of stress axis rotation (after Leblond,2002) (� � �).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 96

6.4 Anisotropy at small and intermediate strains

The anisotropy of Toyoura sand at small strains was shown in Figure 6.3a. Although it was seen to beisotropic at small strains, this is not the case in general. Figure 6.32 shows that initial or inherentanisotropy, measured under isotropic stress ( �v/ �h � 1) in terms of the ratio Ev/Eh, varies with grain sizein granular materials, and can be as large as 2.5. This level of anisotropy is modified as the consolidation

97

0.5

0.4

0.3

0.2

0.1

0.0

2922

0

0.0 0.1 0.2 0.3' ' 0.4 0.5 0.6 0.7 0.8

Strength envelopes for�= 0, 22 and 29

(Failure plane horizontal)°

Strength envelopefor = 37 - 75(Failure plane horizontal)

Stress conditions at failure on horizontal plane

α °

σ σ/

τ/σ

75

65

40

68 60

5350

45

37

p'

p

Figure 6.31. Strength envelopes deduced from undrained HCA tests (after Leblond, 2002) (� � �).

σv'/σh'

Ev/

Eh

0

1

2

3

4

Toyoura sandSLB sandTicino sandHime gravelChiba gravelNerima gravel

Inherent anisotropy

Stress system-inducedanisotropy

Perfectly isotropic material

0 1 2

Figure 6.32. Small strain anisotropy in sands and gravels (Tatsuoka et al., 1997).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 97

stress ratio �v/ �h changes; this stress-system induced anisotropy increases the ratio Ev/Eh to 3.5 in agravel under �v/ �h equal to 2.

Cross-anisotropic elastic parameters that characterise soils inside the Y1 zone have been derived bya combination of effective stress triaxial probing tests and shear wave velocity measurements, made onthe same specimen, under the same effective stress conditions (e.g. Jardine et al., 2001). The assumptionis made that the soil is linear elastic and rate independent inside the Y1 zone. One such set of parame-ters for normally consolidated Ham river sand is presented in Figure 6.33 (Jardine et al., 2001). Thisshows that during K0 consolidation:

• E�v/E�h varies between 1.8 and 2.1• Gvh exceeds Ghh by 20 to 25%

Information on the anisotropy of small strain stiffness of dry Ticino sand, measured at different consolidation stress ratios in a calibration chamber using wave velocities, has been taken from Bellottiet al. (1996) and is presented in Figure 6.34.

A compilation of data on the anisotropy of small strain stiffness in natural and reconstituted soft andstiff clays is presented in Figure 6.35 in terms of the ratio Ghv/Ghh or Gvh/Ghh versus consolidation stressratio �v/ �h. In the mottled facies of the Bothkennar Clay, the fabric of which has been renderedisotropic by bioturbation, the ratio is unity in an unconfined state. In the bedded facies the ratio is 0.8in an unconfined state, consistent with its anisotropic fabric. Jamiolkowski et al. (1994) found Gvh/Ghhratios under isotropic stress conditions of 0.7 and about 0.65 for the natural Pisa & Panigalia claysrespectively. In situ data for the stiff and very stiff London and Gault Clays is very similar, althoughunder isotropic stress conditions the Gault shows a lower ratio for Gvh/Ghh. Nash et al. (1999) foundratios of 0.7–1.9 over the range of void ratio tested for the one-dimensionally reconstituted Gault clayunder isotropic stress.

One of the most comprehensive data sets on stiffness anisotropy at intermediate strains has been pres-ented by Zdravkovic & Jardine (2000), from whom Figures 6.36, 6.37 and 6.38 have been taken.

98

0 50 100 150 200 250

p' (kPa)

0

100

200

300

400

500

600

Sti f

fne s

s(M

Pa )

GhhGvhE'hE'v

Figure 6.33. Variation of stiffness components with p� in dense Ham river sand during K0 consolidation, OCR � 1(Jardine et al., 2001).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 98

99

0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3

Consolidation stress ratio K = σ 'h / σ 'v

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Mo

du

lus

ratio

Gvh / Ev

Ghh / Eh

Ghh / Gvh

Eh / Ev

G/E isotropic medium1.2 K0.251.2 K0.25

DR = 40 - 45 %σ 'h = 100 kPa

Figure 6.34. Small strain stiffness anisotropy in Ticino sand (Bellotti et al., 1986).

σv'/σh'0 1 2 3

Gvh

/Ghh

or

Ghv

/Ghh

0.0

0.5

1.0

Bothkennar in situ (Gvh)

Bothkennar intact (Ghv)Bothkennar intact (Gvh)

London Clay in situ (Gvh)London Clay intact (Gvh)London Clay reconstituted (Ghv)Gault Clay in situ (Gvh)Gault Clay intact (Ghv)Gault Clay reconstituted (Ghv)Pisa Clay intact (Gvh)Panigaglia Clay intact (Gvh)

Figure 6.35. Small strain stiffness anisotropy in natural and reconstituted clays.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 99

Figure 6.36 shows the decay curves for octahedral shear stiffness during undrained shear of the HPF4silt with different principal stress directions and b � 0.5. Stiffnesses for M0 to M70 are, in fact, similarand much softer than for initial unloading. The effect of b on stiffness characteristics is illustrated inFigure 6.37, in which octahedral shear stiffness is compared for compression with b � 0, 0.3 and 0.5and for extension with b � 1 and 0.5. Figure 6.38 compares the stiffnesses determined from uniaxialand torsional shear stages of drained tests on the silt at an OCR of 1.3.

100

140,000

M0M0

shearing stages

M15M15M30

M30M45

M45M70

M70

M90

A

p' (kPa)

t (kP

a)

Goc

t (kP

a)

M90TE

120,000

100,000

80,000

60,000

40,000

20,000

0.001 0.01 0.1

εd (%)

1 100

Figure 6.36. Torsional shear stiffness in HPF4 silt (Zdravkovic & Jardine, 2000).

εs (%)

0.001 0.01 0.1 1 10

Eu

(kP

a)

0

100000

200000

300000

400000

HCA compression, b=0.3HCA compression, b=0.5HCA extension, b=0.5Triaxial compressionTriaxial extension

Figure 6.37. Undrained Young’s modulus for different combinations of b and � in HPF4 silt (Zdravkovic, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 100

6.5 Concluding remarks

Anisotropy of undrained strength can result from the anisotropy of volume change characteristics, as inloose sands, from anisotropy of stiffness, as in stiff clays, from anisotropy of effective stress strengthparameters, as in dense sands or in stiff fissured or jointed clays, in which the discontinuities have adominant orientation.

The anisotropy of natural soils in situ is determined by: the inherent anisotropy of the soil, which canbe considered to be that resulting from the structure of the material, i.e. its fabric and bonding at all lev-els, and can be measured with the soil in an unconfined state; and the in situ stress state, which is likelyto be anisotropic and induces additional anisotropy.

As the fabric features that can influence anisotropy occur at different scales, so will anisotropy bescale dependent. The level of anisotropy and its pattern will vary. Anisotropy at small strains bears norelation to anisotropy at large strains. Because anisotropy is induced by strains, sampling can modifyinherent anisotropy (Hight, 2001).

Examples of the influence of small strain and large strain anisotropy are presented by Hight andLeroueil (2002*). These confirm that efforts should be made to assess the likely level and pattern ofanisotropy and its subsequent influence on performance.

To take into account the effects of anisotropy, it was suggested by Hight & Higgins (1994) that thescheme for yielding introduced by Jardine (1992) be extended as shown in Figure 6.39. As indicated, the curves Y1, Y2 and Y3 decrease in size as the principal stress axis rotates from the vertical directionto the horizontal one in K0 consolidated soil.

7 BEHAVIOUR OF INTERMEDIATE SOILS

Section 5 has considered the behaviour of clean sands. In nature, sands often contain a component offines which may have been included at the time of deposition or may have become incorporated subse-quently. The fines may have a profound effect on behaviour, depending on the quantity, shape, plastic-ity and distribution of the fines, as well as on the grading of the host sand. These effects are illustratedin this Section and are used to illustrate the transition from “sand behaviour” to “clay behaviour”. Thesection also reviews the behaviour of silts and an extreme form of fines in sand, namely the presence ofmica in micaceous sands.

7.1 The effect of clay fines on permeability and compressibility

Perhaps the most important effect of the addition of clay fines to a sand is on its permeability and chang-ing its relevant behaviour from drained to partially drained or even to undrained. Figure 7.1 illustrates

101

εd (%)

0.001 0.01 0.1 1 10

Gz�

, Ghh

, E' (

kPa)

0

50000

100000

150000

200000

εd h'

εd-E

-E

v'

εd-Gz�

εd-Ghh

Figure 6.38. Uniaxial and torsional shear stiffness of K0 consolidated HPF4 silt at OCR � 1.3 (Zdravkovic, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 101

102

αo

0

90

Y2

Y3

p'/p'e

t/p'e

Bodin

ung surface

Y3

Y2Y1

Y3

p'/p'e

e

Projection on t/p’e – p’/p’e plane

't/p

Figure 6.39. Framework for soil behaviour extended to cater for principal stress directions.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 102

how the addition of less than 5% of clay can reduce permeability by more than 3 orders of magnitude.It is apparent that the effect depends on the activity of the clay. The effect of these changes in perme-ability on in situ behaviour is discussed by Hight & Leroueil (2002*).

The effect of clay fines on compressibility is, by contrast, generally limited. Hight et al. (1994) reportthat the compressibility of Ham river sand/kaolin mixtures was dominated by the compressibility of thesand for clay contents up to 20%. With a clay fraction of 20% compressibility was only increased by 1.3to 1.4 times that of loose sand, although the compressibility of normally consolidated kaolin was 20times that of loose sand over a stress range of 300–400 kPa.

7.2 The effect of clay fines on the behaviour of sand in shear

The effect of clay fines on the behaviour of sand was investigated by Georgiannou et al. (1990, 1991a,and 1991b) and summarised by Hight et al. (1994) in connection with the foundation soils at the site ofthe Gullfaks ‘C’ gravity platform in the Norwegian sector of the North Sea.

Both the natural clayey sands and a model clayey sand, comprising Ham river sand and kaolin, wereinvestigated. In the intact clayey sands from Gullfaks, the clay acted as both a separator at the contactbetween sand grains and as a filler to the voids between sand grains, see Figure 7.2. This fabric sug-gested contemporaneous deposition of the sand and clay, with the clay initially in suspension, ratherthan being washed in subsequently to fill only the voids.

Contemporaneous deposition of sand and clay was modelled by raining sand slowly into suspensionsof the clay component in distilled water. This revealed a strong dependence of the settled bed height onthe clay content of the suspension, for a given volume of rained sand. Increasing bed height, which cor-responds to a looser granular component, accompanied increases in clay concentration. In a parallelstudy, the ability of the clay to separate (or loosen) the sand was shown to increase with the activity ofthe clay and the angularity of the sand grains, for a given clay concentration.

The main features of the behaviour in undrained triaxial compression and extension of K0-normallyconsolidated clayey sands, formed in this way, are summarised below, referring to Figures 7.3 to 7.6:

• The addition of relatively small quantities of clay introduces pronounced undrained brittleness in tri-axial compression; this is illustrated in Figure 7.3 for samples having approximately the same gran-ular void ratio, eg

103

Figure 7.1. Effect of clay fines on the permeability of sand (Hight et al., 1994).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 103

104

Figure 7.2. Fabric of clayey sands from Gullfaks “C” site, North Sea.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 104

• Strains to phase transformation also increase with clay content (Figure 7.3)• In undrained triaxial extension, the increased tendency for contraction with the addition of clay leads

to complete liquefaction of the sedimented model clayey sand having clay contents of 3.5 and 7.5%• At similar clay contents, the effects of reductions in eg after sedimentation are to reduce undrained

brittleness and to reduce the risk of liquefaction in triaxial extension (Figure 7.4)

105

Figure 7.3. Undrained triaxial compression and extension tests on K0 normally consolidated loose sand clayeysand (sedimented HRS and kaolin) (Hight et al., 1994).

Figure 7.4. Undrained triaxial compression and extension tests on K0 normally consolidated clayey sands withvarying granular void ratio at similar clay content (Hight et al., 1994).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 105

106

100 200 300 400 500

(σa’ + σr’)/2 (kPa)

200

100

0

-100

(σa

-σr)/

2 (k

Pa)

30

620

7

7.6

8

10Ham river sand pluviated into kaolin suspensionsexcept 20% and 30% (wet slurry)

0Clay content (%)

Figure 7.5. Effective stress paths for undrained triaxial compression on K0 normally consolidated clayey sandswith varying clay content (Hight et al., 1994).

Clay content (%)

0 5 10 15 20 25 30

Bitt

lene

ss in

dex,

I B (

%)

0

20

40

60

80

100

Triaxial extensionb=1, α=90o

Triaxial compressionb=0, α=0o

prepared by wet slurry method

Figure 7.6. Undrained brittleness in sand sedimented into kaolin suspension.

100 200 300 400 500

200

100

0

-100

(σa’ + σr’)/2 (kPa)

(σa

-σ r

)/2

(kP

a)

Ham river sand pluviated into 7-7.8% kaolin suspensioneg= 0.78 - 0.80

OCR = 1.0

1.3

2.0

2.0

1.3

OCR = 1.0

Figure 7.7. Effects of stress history in clayey sand(Georgiannou et al., 1990).

100 200 300 400 500

(σa’ + σr’)/2 (kPa)

200

100

0

-100

(σa

-σr)/

2 (k

Pa) OCR = 1.0

1.3

2.04.0

Figure 7.8. Effects of stress history in pluviated Hamriver sand (Ovando-Shelley, 1986).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 106

• The maximum undrained brittleness in triaxial compression occurs at a clay content of 10%(Figures 7.5 and 7.6); for higher clay contents, undrained brittleness reduces and, at a clay content of30%, the material assumes the behaviour of normally consolidated clay.

The effects of stress history on the behaviour of the model clayey sand are illustrated in Figure 7.7.Overconsolidation reduces undrained brittleness in triaxial compression but does not prevent liquefac-tion in undrained triaxial extension. The effects of stress history on loose sedimented clean sand, involv-ing K0 consolidation and swelling, are included as Figure 7.8 to demonstrate an essential differencebetween clean and clayey sands. Overconsolidated clean sands show gross yielding before reaching thenormalised effective stress paths of the normally consolidated sands, i.e. the Y3 yield surface shrinks onunloading. In clayey sands, the presence of only small quantities of clay fixes the structure giving thesoil a memory of its stress history.

A problem related to the characterisation of clayey sands discussed by Hight & Leroueil (2002*)concerns their particular range of permeability, which can lead to a switch between undrained anddrained behaviour. An illustration of the effects of the switch is presented in Figure 7.9 which comparesthe drained and undrained behaviour of the model clayey sand at OCRs of 1 and 1.3.

107

00

200

200

400 500[�'o - �'r] /2 [kPa]

[�o

- �

T] /

2 [k

Pa]

DCR = 1,3

1,0

0 0.5

200

100

300

400

1.0

[�o

- �

r] [k

Pa]

Axial strain, �a (%)

Drained

Undrained

Figure 7.9. Comparison of drained and undrained behaviour of clayey sand in triaxial compression (Georgiannouet al., 1990).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 107

7.3 The effect of mica on the behaviour of sand

The behaviour of micaceous sands was studied in connection with flow slides which occurred duringconstruction of river training works for the Jamuna Bridge in Bangladesh (Hight et al., 1999). Detailsare given by Hight & Leroueil (2002*), who use it as a case history to illustrate the importance ofanisotropy and collapse potential in sands.

Two batches (A and B) of the natural micaceous sand, having the gradings shown in Figure 7.10,were tested, together with mixtures of a model sand and an industrial mica (MF60), which sat within thegrading envelopes shown in Figure 7.10. The mica in the natural sand comprised thin sand-sized platesof biotite, derived from the crystalline rocks of the Himalayas; the quantity of mica, its distribution andits orientation varied. Grain counting indicated mica contents of 5–10% by number of grains. SPTs atthe site had indicated relative densities between 40 and 60%.

7.3.1 Effect of mica content on void ratioIt had been demonstrated in the classic study by Gilboy (1928) that the addition of sand sized mica to adumped sand increased its porosity dramatically. We repeated Gilboy’s experiments, using the host sandand MF 60 mica. The results are shown in Figure 7.11, in terms of the void ratio measured in the dumped

108

Figure 7.10. Particle size distributions of micaceous sands (Hight et al., 1999).

Percentage of fines

0 5 10 15 20

Voi

d ra

tio

0

1

2

Sand sized micaAngular siltKaolin

Figure 7.11. Effects of fines on the void ratio of a sand (Hight et al., 1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 108

sand as the mica content by weight was increased from 0% to 20%. For comparison, Figure 7.11 alsoshows the void ratio in the same sand when increasing percentages of angular silt were added, and in asimilar sand, Ham river sand, when sedimented into kaolin suspensions of increasing clay concentra-tions. In contrast to the effects of adding silt or kaolin to the sand, the effect of adding mica is to increasethe void ratio monotonically with mica content. When dumped and under low stress, the micaceoussand can be expected, therefore, to be in a loose state.

7.3.2 Effect of mica on volume change characteristics and collapse potentialSamples of the model sand-mica mix, with different mica contents by weight, have been tested in constant volume simple shear (CVDSS), using the Geonor DSS device (Bjerrum & Landva, 1966). Thesamples were formed by dumping the mix through a trap door into the reinforced membrane.Figure 7.12a compares the behaviour of a clean sand and the same sand with just 1% of mica added. Theclean sand, although loose, is ductile, a tendency to dilate at large strains causing the effective stress

109

Shear strain (%)

0 4 8 12 16 20

0 % mica

1 % mica

Vertical effective stress (kPa)

0 20 40 60 80 100 120

Hor

izon

tal s

hear

str

ess

(kP

a)

0

20

40

60

80

100

0 % micae = 0.7

1 % micae = 0.74

Shear strain (%)

0 5 10 15 20

Vertical effective stress (kPa)

0 40 80 120

Hor

izon

tal s

hear

str

ess

(kP

a)

0

20

40

60

80

0% mica

2040

105

2.51

0

20

4010

5

2.51

Shear strain (%)

0 10 20

% mica0 (e = 0.76)

5 (e = 0.76)

5 (e = 0.91)

Vertical effective stress (kPa)

0 20 40 60 80 100 120

Hor

izon

tal s

hear

str

ess

(kP

a)

0

20

40

60

80

100

% mica0 (e = 0.76)

5 (e = 0.76)

5 (e = 0.91)

(a)

(b)

(c)

Figure 7.12. Effects of mica on the behaviour of sand in constant volume simple shear (Hight et al., 1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 109

path to climb up the failure line. In stark contrast, the sand with 1% mica is brittle and shows a poten-tial to collapse. The addition of just 1% of mica by weight has suppressed almost completely the tendency to dilate. It is important to note that, because of the aspect ratio of the mica plates (of approx-imately 50:1) compared to the rotund sand particles, 1% of mica by weight is approximately equivalentto 25% of mica by number of grains.

Figure 7.12b shows the behaviour observed in a series of CVDSS tests on dumped sand samples having mica contents ranging between 0% and 40% by weight. This illustrates that it is relatively smallquantities of mica that are critical, in terms of brittleness and collapse potential. Higher percentages ofmica increase the stability of the mix, despite the increase in void ratio. At 40% mica, the behaviour issimilar to that of a normally consolidated unstructured clay, with no tendency to dilate or collapse;behaviour at large strains is clearly dominated by the mica, which gives rise to a lower � (23° cf 33°).In this latter respect, the addition of mica is similar to the addition of kaolin described above.

It might be argued that the effect of the mica on the behaviour of the sand is simply a reflection of itseffect on void ratio. That this is not the case is demonstrated by the data in Figure 7.12c. This com-pares the results of CVDSS tests on dumped sand samples having mica contents of 0% and 5% and voidratios of 0.76 and 0.91, and a second sample with 5% mica, brought to a similar void ratio as the cleansand. The two samples with 5% mica behave in a similar way, despite their differences in void ratio. Theclean sand and sand with 5% mica at similar void ratio behave quite differently, with the mica again sup-pressing dilation.

7.3.3 Anisotropy of undrained strength in micaceous sandsA series of undrained triaxial compression and extension tests has been run on the model sand-micamix, using air-pluviated samples with increasing mica contents, and K0 consolidated before undrainedshear. The results are shown in Figure 7.13. In triaxial compression, the addition of mica actuallyincreases the initial stability of the structure, reducing undrained brittleness; this is in direct contrast tothe effects of adding kaolin to Ham river sand described in Section 7.2. In undrained triaxial extension,small mica contents increase brittleness and give the sand an even greater potential to collapse; furtherincreases in mica content suppress dilation at large strains in extension and begin to stabilise the mate-rial. With small mica contents, the sand is extremely weak when loaded in extension.

This huge difference in strength between undrained triaxial compression and extension was also foundin similar tests on the natural material and was seen to persist at relative densities of 55%. Figure 7.14ashows the results of undrained triaxial compression and extension tests on samples from Batch A, prepared by dry spooning at relative densities of 58% and 55%, respectively. In compression, the mate-rial is ductile and has an undrained strength in excess of 180 kPa. In undrained extension, the materialis brittle and has a strength of only 10 kPa. Figure 7.14b provides a similar comparison, but betweensamples prepared from Batch B by moist tamping at relative densities of 25 to 27%; again strengths inextension are extremely low and there is no tendency to collapse in compression even at these low relative densities.

110

(σa' + σr')/2 (kPa)

0 25 50 75 100( σa

- σ r)/

2 (k

Pa)

-25

0

25

50

mica content (%)

-30 -10 10 30-20 0 20

-25

0

25

500

1

2.5

10

20

40

Axial strain (%)

Figure 7.13. Effects of mica on the behaviour of sand in undrained triaxial compression and extension (Hight et al.,1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 110

7.3.4 Effect of fabric in micaceous sandsNot only does the mica content vary in the natural materials, but, as noted above, so does its distribu-tion and orientation within the sand. The method of preparing specimens of the micaceous sands fortesting was varied, therefore, as this changed the orientation and distribution of the mica. Thus, in sam-ples prepared by air pluviation or dry spooning, the mica was relatively uniformly distributed andaligned horizontally. In samples prepared by moist tamping, the mica distribution was less uniform andits orientation more random; in addition, following the observations of Jang and Frost (1998), the localporosity would have shown a greater variation, with the presence of larger and more collapsible pores.

Figure 7.15 compares the behaviour in undrained triaxial compression of samples prepared by dryspooning and moist tamping. Figure 7.15a shows data for Batch A and Figure 7.15b for Batch B of thematerial. The effect of preparation method is critical. A test specimen of material from Batch A formedby dry spooning at a relative density of 40% dilates strongly while a specimen formed by moist tamp-ing at a similar relative density collapses. The reverse pattern is observed in Batch B, with the speci-mens formed by dry spooning showing the potential to collapse. The effect of preparation method ofBatch B in constant volume simple shear is illustrated in Figure 7.16. Again, the effect of fabric is mostpronounced, although both batches show contractant behaviour when samples are prepared by moisttamping.

At similar relative density, a micaceous sand may, therefore, be ductile or brittle, with a potential tocollapse, depending on its fabric, i.e. on its method of placement and grading, and, therefore, on the dis-tribution and orientation of the mica.

7.3.5 Collapse surface in micaceous sandsThe concept of a collapse surface and its key features were described in Sections 5 and 6. Some of thesefeatures in the artificial micaceous sand with 2.5% mica are illustrated by the test results shown in

111

Axial strain (%)

-20 -10 0 10 20

Dr (%)

58.3

(σa'+σ

r')/2 (kPa)

0 100 200 300

(σa-σ

r)/2

(kP

a)

-100

0

100

200 Dr (%)58.3

54.6 54.6

Figure 7.14a. CAU triaxial compression and extension tests on micaceous sand. Batch A, dry spooned (Hight et al.,1999).

(σa'+σ

r')/2 (kPa)

0 50 100 150

(σa-σ

r)/2

(kP

a)

-50

0

50

100

24.9

Dr (%)

26.9

26.8

Axial Strain (%)-10 0 10 20

(σa-σ

r)/2

(kP

a)

-50

0

50

100

Dr (%)

26.9

24.9

26.8

Figure 7.14b. CAU triaxial compression and extension tests on micaceous sand. Batch B, moist tamped (Hight et al., 1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 111

112

(a) Batch A

(σa-

σ r)/2

(kP

a)

0

40

80

120

Dr (%)39.6

37.1

(σa'+σr')/2 (kPa)(b) Batch B

0 40 80 120 160

(σa-

σ r)/2

(kP

a)

0

40

80

120

Moist tampedDry spooned

24.9

Dr (%)31.2

26.336.7

27.8

Collapse

26.9

Figure 7.15. CAU triaxial compression tests on micaceous sand. Effect of preparation method and batch (Hightet al., 1999).

Hor

izon

tal s

hear

str

ess

(kP

a)

0

20

40

60

Moist tampedDry Spooned

Dr (%)44.7

30.4

43.7

31.4

Figure 7.16. Constant volume simple shear tests on micaceous sand. Effect of preparation method, Batch B (Hightet al., 1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 112

Figure 7.17. The samples were formed by moist tamping and, as can be seen, moist tamping producessamples that collapse under triaxial compression; this allows the features of the collapse surface to beinvestigated in relatively simple tests. The fact that the effective stress path from the compression testdefines part of the collapse surface is demonstrated by a second test on an identical moist tamped sam-ple, which was held at constant t (�( a � r)/2) and swelled back under stress controlled conditions.The stress path followed and the response on crossing the collapse surface are shown in Figure 7.17. Thedevelopment of strain accelerated abruptly on crossing the surface. Drained conditions were maintainedand the stress path continued at constant t until the full friction angle was mobilised. Had drainage atthe time of collapse been impeded, then the effective stress path would have followed the collapse sur-face as indicated in Figure 7.17.

On the basis of the results presented previously, the collapse surface for pluviated micaceous sandswould be restricted to high values of �, and would be particularly low at � of 90°, i.e. for extension loading.

It is evident from the data that has been presented that mica can introduce a collapse potential for cer-tain directions of loading at values of relative density at which collapse would not normally be expected.In the same vein Yoshimine et al. (1998) present data (refer Figure 7.18) that shows that with increasingfines content the relative density at which there can be zero residual strength increases.

7.4 The effect of silt fines on sand behaviour

The effects of fines described to date relate to clay and to sand size mica plates. Adding rotund silt par-ticles has a different effect, tending to stabilise loose sand. This is evident from data presented byKuerbis et al. (1988) for undrained triaxial compression and extension tests on isotropically consoli-dated sand with increasing proportions of silt (Figure 7.19). The silty sands were formed by slurry depo-sition and the silts presumably occupied the void space rather than increasing the separation between

113

(σa-σ

r)/2

(kP

a)

0

25

50

e=0.934

e=0.917

Undrained triaxial compression

Collapse

Constant t

Undrained

Drained

(σa'+σ

r')/2 (kPa)

0 25 50 75 100

Axia

l str

ain

(%

)

0

10

20

Figure 7.17. Collapse in sands with 2.5% mica prepared by moist tamping (Hight et al., 1999).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 113

sand grains. This effect of silt has been confirmed for K0 consolidated sands. Figure 7.20 compares thebehaviour in undrained triaxial compression of a clean sand and the same sand with 1% mica, 2.5%mica and 2.5% angular silt, all formed by slurry deposition. As illustrated by Figure 7.21, the effect offines on the stability of the structure will depend on the shape, location and quantity of the fines.

114

Fines content (%)

0 4 8 12 16

Rel

ativ

e de

nsity

for

esse

ntia

lly z

ero

resi

dual

str

engt

h,D

r (%

) at

I B=

1.0

10

30

50

70

0

20

40

60Triaxial compressionTriaxial extensionSimple shear

Figure 7.18. Relative density for essentially zero residual strength (Yoshimine & Ishihara, 1998).

Figure 7.19. Undrained triaxial compression and extension tests on isotropically consolidated silty sands (Kuerbiset al., 1988).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 114

7.5 Some features of the behaviour of silt

Silts are often regarded as difficult or intermediate soils. This stems in part from the difficulty in sam-pling and handling them. Unlike sands, and because of their lower permeability, silts are able to sustaina small suction, giving them some coherence. Being at a low effective stress when unconfined they willnaturally show a tendency to dilate. As with sands, silts can be deposited at a range of void ratios andhave a multiplicity of normal consolidation lines. Again, in common with sands and glacial tills, theirbehaviour is not uniquely related to void ratio and stress history, but depends on initial porosity, stresslevel and stress history. The effect of initial porosity of HPF4 silt, which has the grading shown inFigure 7.22, is evident from data from undrained triaxial compression tests on isotropically consoli-dated samples in Figure 7.23.

An investigation of the same silt in a dense state (e0 � 0.65, Dr � 88%) is reported by Zdravkovicand Jardine (2000). Figure 7.24 defines the behaviour of the silt after K0 consolidation and swelling ofsamples prepared by pluviation. The silt has no memory. It is highly dilatant at large strains. Themobilised friction angle at phase transformation is 32° in compression and 25° in extension. The max-imum mobilised friction angle was 37° in compression and 30° in extension.

115

(σa'+σr')/2 (kPa)0 25 50 75 100

( σa-

σ r)/2

(kP

a)

0

25

50

0 % mica 0.792

1 % mica 0.829

2.5 % mica 0.893

2.5 % HPF4 silt 0.780

e

2.5% silt

0% mica

2.5% mica

1% mica

Axial strain (%)0 5 10 15 20 25 30

(σa-

σ r)/2

(kP

a)

0

25

50

0 % mica 0.792

1 % mica 0.829

2.5 % mica 0.893

2.5 % HPF4 silt 0.780

e

Figure 7.20. Undrained triaxial compression of sands with 0% and 2.5% mica and 2.5% HPF4 silt.

(c) (d)

(a) (b)

Figure 7.21. Importance of shape and location of additives in sand.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 115

8 LOCALIZATION AND BEHAVIOUR AT LARGE DEFORMATIONS

8.1 Generalities on post-failure and localization

Critical state soil mechanics (CSSM) described in Section 2 considers that, during shearing, the soildeforms homogeneously and reaches critical state at large strains or deformations, whether the speci-men is normally or overconsolidated. When specimens of normally consolidated or slightly overcon-solidated clays, or loose cohesionless materials are subjected to triaxial compression tests, they indeeddeform in an essentially homogeneous manner up to large strains. If there are some end restraints, thespecimens will take a barrel shape but there is no formation of well-defined shear zones. Conditionsreached at large strains are then in accordance with CSSM. On the other hand, overconsolidated claysor dense cohesionless materials subjected to triaxial compression tests show the formation of one or

116

0

10

20

30

40

50

60

70

80

90

100

0.001 0.01 0.1 1

Grain Size (mm)

% p

assi

ng

Silt AHam River Sand

Figure 7.22. Grain size distribution of HPF4 silt compared to Ham river sand.

0 100 200 300 400 500(σa'+σr')/2 (kPa)

0

100

200

300

400

(σa-

σ r)/2

(kP

a)

42.6%

39.7% 37.7%

0 4 8 12Axial strain (%)

0

100

200

300

400

(σa-

σ r)/2

(kP

a)

42.6%

39.7%

37.7%

Figure 7.23. Effect of initial porosity on behaviour of isotropically consolidated HPF4 silt in triaxial compression(Ovando-Shelley, 1986).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 116

117

Figure 7.24. Effective stress paths for HPF4 silt in undrained triaxial compression and extension (Zdravkovic &Jardine, 2000).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 117

several well defined shear bands (there is localization), followed by essentially rigid bodies sliding overothers (Viggiani et al., 1993; Tillard-Ngan et al., 1993; Desrues et al., 1996).

The phenomenon of localization is illustrated in Figure 8.1 using the results of an undrained triaxialtest carried out on stiff Todi clay. The instrumentation for this test included, in addition to an externalaxial displacement transducer and a pore pressure transducer at the base, local strain transducersdirectly on the soil specimen and a local pore pressure probe (see the schematic specimen in Figure 8.1).For the test presented in Figure 8.1, the failure surface passed outside the local strain transducers andclose to the pore pressure probe. Burland (1990) made several remarks in relation to the results shownin Figure 8.1, among which: (a) prior to peak strength the local strains are less than the overall strains,as was observed by many other authors; (b) after peak strength is reached the local axial strains decreaseas a result of the unloading process; and (c) localization essentially coincides with peak strength.

When displacement along the shear surface progresses, depending on the shape of soil particles,there may be an orientation of particles in the direction of movement and a progressive decrease in shearstrength towards the residual strength.

From what has been previously said, three different levels of strength can generally be defined forsoils: (a) the peak strength of the soil in the pre-yield (or overconsolidated) range, which reflects thetrue friction angle and cohesion as expressed by Hvorslev (1937; see Section 2.1) and, in many naturalgeomaterials, bonding between particles or aggregates defined here as microstructure; (b) the criticalstate strength envelope associated with large and homogeneous deformations or post-rupture strength;and finally, (c) the residual strength envelope associated with oriented particles and very large dis-placements along a slip surface. As shown in Figure 8.2, these three levels of strength were alreadyidentified by Tiedemann in 1937.

8.2 Localization in cohesionless materials

Yoshida & Tatsuoka (1997) investigated the deformation characteristics of shear bands in cohesionlessmaterials and their relation to the physical characteristics of particles. These authors performed a seriesof plane strain compression tests on 12 poorly graded materials with uniformity coefficient Cu between1.2 and 2.13 and mean particle diameter D50 between 0.174 and 6.80 mm. All the tests were performedon dense specimens so that localization could occur. The shear deformation of the shear band necessaryafter peak to reach residual or steady state conditions, (us*)res, and the shear band thickness were

118

Figure 8.1. Unconsolidated undrained triaxial test on Todi clay showing post-rupture behaviour (from Burland,1990).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 118

determined; they are plotted in Figure 8.3 as a function of D50. It can be seen that both shear displace-ment (us*)res and shear band thickness increase as D50 increases. Figures 8.3a and b also show that theseshear band characteristics decrease as the confining pressure increases. Yoshida & Tatsuoka (1997) alsoindicate that shear band thickness becomes larger as particles are rounder or less crushable.

119

Peak strength (intact specimens) Peak strength (remoulded specimens) Residual strength (intact specimens) Residual strength (remoulded specimens)

2.0

1.5

1.0

0.5

00 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

σ (kg/cm )2

τ (k

g/cm

)2

Figure 8.2. Failure envelopes of Weser-Elbe clay (after Tiedemann, 1937).

Figure 8.3. Influence of particle diameter D50 on shear band thickness (a) and (u*s)res (b) (from Yoshida & Tatsuoka,1997).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 119

Using computer tomography (X-ray scanner), Desrues et al. (1996) observed the local evolution ofsoil density inside triaxial specimens of sand subjected to constant effective confining pressure com-pression tests, and in particular the development of localization. Figure 8.4 shows the evolution of bothaverage and local (in the shear band) void ratios with global axial strain. Several conclusions can bedrawn: (a) on the basis of average void ratios observed at large deformation, which are equal to 0.85 and0.77 for the loose and dense specimens respectively, the critical state concept would not apply; (b) thevoid ratio in the shear bands observed in the initially dense specimens (mostly the black dots) startsincreasing at some stage to reach the value obtained at large strains in the loose specimens, indicatingthat the principle of critical state applies when conditions in the shear bands are considered; when spec-imens are contractant, the average void ratio is representative of critical state conditions. It is worth not-ing that Casagrande & Watson (1938) stated these points more than sixty years ago. An importantconclusion of Desrues et al’s study is that the concept of critical state applies in dense sands, but locally;the shear strength is then characterized by the critical state friction angle, ��cs.

8.3 Post-rupture strength in microstructured soils

On the basis of direct shear tests performed on intact stiff Todi clay, Calabresi (1980) distinguished twostages of post-peak strength reduction: the first one occurs just after peak, at a relative displacement ofless than 1 mm; the second takes place gradually as displacement increases. These two stages were con-firmed for several stiff clays by Burland (1990) and Burland et al. (1996). As shown in Figure 8.1, afterpeak, the curve of deviator force versus notional overall strain falls abruptly to a kind of plateau. Thisplateau was termed “post-rupture strength”. These authors also showed that this post-rupture strength lieson or very close to the critical state or fully softened strength envelope. The strength reduction from peakto post-rupture thus mostly corresponds to the breakage of the bonds between particles or aggregates.

These results are supported by laboratory and field observations made by Calabresi & Manfredini(1973) and Skempton (1977). Measuring the strength along fissures and joints along which no dis-placement had occurred previously, these authors measured strengths very close to critical state strength(see Section 12.1 and Figures 12.2 and 12.3).

However, if the soil contains stiff aggregates or lithorelicts, the large deformation strength envelopeobtained in triaxial testing of a natural soil can be above the critical state strength envelope defined onthe same material, reconstituted or destructured by compression beyond the limit state curve. This hasbeen shown by Leroueil et al. (1997) for different clays, sensitive eastern Canada clays and softened clayshales from southern Italy. As shown in Figure 8.5 for the Saint-Jean-Vianney clay, the large deformation

120

0 0.10 0.20 0.30 0.40 0.50Axial strain (global)

1.00

0.90

0.80

0.70

0.60

0.50

Voi

d ra

tio (g

loba

l or l

ocal

)

rfdt3 rfdt4 rfdt6 rfdt7 rfdt8 rflt2 rflt1

Local GlobalVoid ratio evolution

(D = 0.32 mm; C = 1.7) σ' = 60 kPa

50 u

c

Figure 8.4. Global and local evolution of the void ratio in loose and dense Hostun RF sand specimens subjected totriaxial compression test (from Desrues et al., 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 120

friction angle of the reconstituted soil is in the order of 30° whereas the large deformation friction angleof the natural and overconsolidated clay is about 44°. This was attributed to the fact that sensitive claysfrom eastern Canada are made up of aggregates that are stiff and extremely angular. Soccodato (2002*)observed a similar behaviour in Fucino clay.

8.4 Residual strength

Residual conditions, which are reached after large displacements along a slip surface, are characterizedby a friction angle, ��r, and zero cohesion. In soils made of rotund particles, shearing does not introduceany preferential orientation of the particles and ��r is equal to or only slightly smaller than ��cs (turbulentshear according to Lupini et al., 1981); in soils made up of platy particles, shearing induces particle orientation in the direction of shearing and ��r is significantly smaller than ��cs (sliding shear accordingto Lupini et al., 1981). Values of ��r in clayey materials are given in Section 16.6 and Figures 16.4 and 16.5.

The behaviour of soils containing both rotund and platy particles has been studied by Lupini et al.(1981). As shown in Figure 8.6, �/ �n values obtained at residual conditions, and thus ��r values, are

121

300

200

100

00 100 200 300 400 500 600

' = 44.4˚

' = 30.5˚

(σ' + σ' )/2 (kPa)a r

(σ'

- σ '

)/2

(kP

a)a

r Reconstituted soil CSL

Overconsolidated Normally consolidated

Natural soil

Saint-Jean-Vianney clay

I = 8 - 10%; σ

φ φ

' ~ 1000 kPa p p

Figure 8.5. Large deformation strength of the natural and reconstituted Saint-Jean-Vianney clay (from Saihi et al.,2002).

Rolling shear Transitional Sliding shear

0

0.2

0 20 40 60 80 100

0.4

0.6

0.8

Clay fraction (%)

Fric

tion

coef

fici

ent τ

/ σ ' n

'r

n.c. cr'orφ φ

φ

'

Figure 8.6. Ring shear tests on sand-bentonite mixtures (after Lupini et al., 1981, and Skempton, 1985).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 121

high when the percentage of platy particles is small (turbulent shear dominant), and small when the per-centage is high (sliding shear dominant). In between, there is a range of percentage in which the behav-iour is transitional: there are rotund particles that impede the development of a perfectly smooth shearsurface, with a resulting residual friction angle intermediate between those of the soils that would bemade up of rotund or platy particles. The development of a perfectly smooth shear surface may also beimpeded in clayey soils, as clay shales, that contain hard lithorelicts. The residual strength envelope ofthe natural soil may then be above the residual strength envelope of the same reconstituted material.This is shown in Figure 12.5 for the Laviano tectonized clay shale from southern Italy.

8.5 Brittleness

Brittleness is associated with strain softening behaviour, i.e. a reduction in shear strength as strainsincrease. It can be observed in most natural soils: stiff plastic clays and shales with residual strengthsmaller than peak strength (Bishop, 1967); microstructured soils when bonds between particles are bro-ken due to shearing; dense sands during drained shearing; loose sands during undrained loading (Sladenet al., 1985); soils subjected to stress axis rotation during shearing or consolidated under an inclinedmajor principal stress (see Section 6); collapsible unsaturated soils. Also, as indicated in Hight &Leroueil (2002*), brittleness has important practical implications since it controls progressive failureand post-failure phenomena.

Bishop (1967) characterized brittleness by a brittleness index, IB, defined as follows:

(8.1)

where �p and �r are the peak and residual strength defined under the same effective normal stress.However, as indicated by Vaughan & Hamza (1977) and by Chandler (1984), the brittleness index

alone is not sufficient to characterize the susceptibility of a soil to progressive failure or to flow; the rateat which the strength decreases from peak to ultimate strength, and thus the energy necessary to go frompeak to ultimate conditions are also important. D’Elia et al. (1998) proposed a generalised brittlenessindex, IGB, defined as follows:

(8.2)

where �mob is the mobilised shear stress at the considered strain or displacement. IGB thus varies withstrain or displacement from 0 at the peak to a value equal to IB at large strain or displacement(Figure 8.7). In the generalised form proposed by D’Elia et al. (1998) and shown in Equation 8.2, the

IGBp mob

p�

� � �

�, %

IBp r

p�

� � �

�, %

122

Strain or displacement

Shea

r stre

ss

τ

τ

I

I

p

r

GB

GB

0

1

=τ p - τ (ε or ∆I)

τp

=τp

τp

τ r-= IB

Figure 8.7. Brittleness of soils (from D’Elia et al., 1998).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 122

brittleness index is associated with a given stress path or test, and must not be seen as a fundamental orintrinsic characteristic of a soil.

9 INFLUENCE OF STRAIN RATE AND TIME

9.1 One-dimensional (and isotropic) compression

It is well known that when a soil specimen is loaded under a constant total vertical stress in one-dimensional conditions, it continues to settle after full dissipation of excess pore pressures. This is the secondary consolidation phase usually characterized by the secondary compression indexC�e � �e/� log t. Mesri & Godlewski (1977) and Mesri et al. (1995) showed that C�e is related to thecompression index Cc of the soil and, more precisely, that the ratio C�e/Cc is a constant for a given soil.Figure 9.1 shows typical results. This has been confirmed for a large variety of geotechnical materialsby Mesri and co-workers, and many other researchers. An interesting aspect is that, as indicated in Table 9.1,C�e/Cc remains within a very narrow range for each soil type, varying from 0.01–0.03 for granular soils to0.05–0.07 for peat (Mesri, 1987; Mesri et al., 1995). Coop & Airey (2002*) found 0.013 for carbonatesands. A direct consequence of secondary consolidation is that, when the soil is reloaded, it shows anapparent preconsolidation pressure larger than the previously applied stress, associated with its lowervoid ratio. As schematically shown in Figure 9.2a, after 10 years, the preconsolidation pressure wouldhave increased from �vo to �p10; it would be �p10 000 after 10 000 years. An indirect consequence is that,as shown by Crawford (1964), Bjerrum (1967), Sällfors (1975) and Leroueil et al. (1983a, 1985a), whenthe soil is reloaded slowly, a reduction of the apparent preconsolidation pressure of the soil is observed

123

Figure 9.1. Values of secondary compression index C�e and compression index Cc for Saint-Hilaire clay (fromMesri et al., 1995).

Table 9.1. Viscous parameters for geotechnical materials.

Material C�e/Cc m�

Granular soils, including rockfill 0.02 � 0.01 100�33 2.3�7.2Shale and mudstone 0.03 � 0.01 50�25 4.7�9.6Inorganic clays and silts 0.04 � 0.01 33�20 7.2�12.2Organic clays and silts 0.05 � 0.01 25�17 9.6�14.8Peat and muskeg 0.06 � 0.01 20�14 12.2�17.5

Note: Types of material and C�e/Cc values are from Mesri et al. (1995).

�

� �

′′

/ log � (%)

09031-02[1].qxd 18/Oct/02 12:10 AM Page 123

(dashed compression curve in Figure 9.2), i.e. the apparent preconsolidation pressure is dependent onthe rate of strain during reloading.

Leroueil et al. (1985a) performed a variety of oedometer tests on different natural clays: MultipleStage Loading tests with reloading at the end of primary consolidation or after 24 hrs, Constant Rate of

124

Figure 9.2. Increase of the preconsolidation pressure and development of the limit state curve with secondary con-solidation (after Tavenas & Leroueil, 1977).

Figure 9.3. Effects on one-dimensional compression of Saint-Polycarpe clay of: (a) strain rate; and (b) temperature(after Marques, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 124

Strain (CRS) tests, Controlled Gradient tests and long-term creep tests. They found that the behaviouris controlled by a unique vertical effective stress-vertical strain-vertical strain rate ( �1–�1–�1) [or( �1–e– e·)] relationship. This rheological model, originally proposed by Suklje (1957), was advocated,in particular, by Lowe (1974). It has also been confirmed experimentally by Imai & Tang (1992) forYokohama Bay mud (see also Imai, 1995). It can be described by compression curves deduced fromCRS tests performed at different strain rates, as shown in Figure 9.3a. Leroueil et al. (1985a) alsoshowed that the ( �1–�1–�1) relationship can be described by two curves, one giving the variation of thepreconsolidation pressure �P with strain rate (see footnote 2):

�p � f(�.1) (9.1)

and the other presenting the normalised effective stress-strain curve:

�1/ �p(�1) � g(�1) (9.2)

This means that the ratio between the preconsolidation pressures, or effective stresses at the samestrain, at two different strain rates is a constant regardless of the strain or void ratio considered. This alsomeans that the horizontal distance between two compression curves corresponding to different strainrates is constant in a �1 (or e)–log �1 diagram.

Figures 9.4a and b show the two curves represented by Equations 9.1 and 9.2 for Berthierville clay.As seen in Figure 9.4a for Berthierville clay, and as observed for other clays, Equation 9.1 can generally

125

Figure 9.4. One-dimensional compression of Berthierville clay: (a) preconsolidation pressure as function of strainrate and temperature; and (b) normalized effective stress-strain curve (from Boudali et al., 1994; Kabbaj, 1985).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 125

be approximated by a linear relation in a log �p–log �1 diagram

log �p � A � (1/m�) log �.1 (9.3)

in which A and m� are constants.Leroueil et al. (1983a, 1985a) and Leroueil (1996) showed that the variation of the preconsolidation

pressure with strain rate is very similar for numerous non-organic clays, i.e. 7 to 15% per logarithmcycle. The corresponding m� value is generally between 17 and 35.

As it can be demonstrated that:

m� � 1/(C�e/Cc) � Cc/�e (9.4)

such m� values are equivalent to C�e/Cc values between 0.059 and 0.029, which coincides with theresults obtained by Mesri and co-workers (Table 9.1). Equation 9.4 also implies that the C�e/Cc and thestrain rate (m�) approaches used for describing the viscous behaviour of soils during secondary consol-idation are equivalent.

In a viscous model such as the ( �1–�1–�.1) [or ( �1–e–e·)] model previously described, soil behaviour

is controlled by the same rheological law, regardless of whether consolidation takes place during its pri-mary or secondary phase.

Since the C�e/Cc � constant model seems to apply for a large variety of soils during secondary con-solidation (Table 9.1), the ( �1–�1–�1) model previously described should also be general. Cotecchia(2002*) showed its application to the hard Pappadai clay; Edil & den Haan (1994) showed its validityfor a fibrous peat; tests performed at different rates of loading by De Waal (1986) indicate that it is validfor sand; finally, tests performed on chalk by Ruddy et al. (1989) and on salt rock by Aubertin et al.(1999) also indicate that the model may be valid for soft rocks.

The implications of the effects of strain rate have been described in detail elsewhere (Leroueil et al.,1985a; Leroueil, 1988, 1996) and only the main aspects will be presented here.

Assuming that primary consolidation is completed before 24 hours in conventional multiple stageloading oedometer tests, the strain rate at the end of the loading periods, which can be associated withthe 24-hrs compression curve, is given by the following equation (Leroueil, 1988):

(9.5)

This strain rate is thus on the order of 5 � 10�8s�1 for low compressibility clays and on the order of10�7s�1 for highly compressible clays. This is smaller than the strain rates used in CRS tests that aregenerally between 1.0 and 4.0 � 10�6s�1. As a result, the preconsolidation pressure and the effectivestress at any strain or void ratio measured in CRS tests are larger than those measured in conventional24-hrs oedometer tests (Figure 9.5). A compilation made by Leroueil (1996) for a variety of clays fromdifferent countries indicates a ratio �p CRS/ �p conv typically equal to 1.25.

When a soil specimen is instantaneously loaded, the strain rate is high during the early stages of con-solidation and, consequently, the effective stresses go to compression curves corresponding to thesehigh strain rates. However, with time, consolidation progresses and the strain rate decreases. As a result,the effective stress-strain condition of the soil moves towards compression curves corresponding tolower strain rates. It follows that, in multiple stage loading tests, the effective stress-strain relationshipreally followed consists of a succession of steps as shown in Figure 9.6, very different from the com-pression curve obtained by joining the points obtained at the end of the loading periods.

Even within a consolidating sample, the effective stress-strain relationships followed in differentparts of the sample depend on the local strain rate history. The test results presented by Mesri et al.(1995) can be used for illustrating this point. These authors performed an isotropic compression test onfour 125 mm-long specimens of St-Hilaire clay connected in series. They present in detail the variationwith time of the axial strain of the four sub-elements of the 500 mm-long specimen and the excess porepressures measured in between for the pressure increment from 97 to 138 kPa during the primary con-solidation phase. In these tests, the compression was isotropic, but drainage was one-dimensional. Fromthese data, it was possible to estimate the effective stress-void ratio curve for the different sub-elements.These curves are drawn in Figure 9.7, from I (initial conditions) to P. Also drawn on the figure is a fictitious secondary consolidation phase (PF). It can be seen that the compression curves followed during primary consolidation vary with the position of the sub-element. Near the drainage boundary(sub-element 1), the strain rate is very high during the early stages of consolidation and, as a result, the

�� � ��

� �24

1 72 101h

c

0s

C

e( )

126

09031-02[1].qxd 18/Oct/02 12:10 AM Page 126

effective stress goes to compression curves associated with high strain rates. On the other hand, for thesub-elements that are far from the drainage boundary (sub-elements 3 and 4), the strain rates are muchsmaller during the same period and the effective stress remains close to compression curves associatedwith these strain rates. However, when the soil is approaching the end-of-primary consolidation (point P),the strain rate becomes more uniform in the specimen and the compression curves deduced from thedifferent sub-elements converge. After the end-of-primary consolidation, the entire specimen would be

127

Figure 9.5. Typical comparison between oedometer test curves obtained in CRS and conventional oedometer tests(Batiscan clay, Leroueil et al., 1983b).

Figure 9.6. Multiple stage loading (MSL) oedometer tests.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 127

in secondary consolidation and would settle from P to F. This test allows one to define the complete setof isotaches (lines of equal strain rate or change in void ratio) for the St-Hilaire clay. Imai & Tang (1992)observed similar behaviour in a consolidating clay layer consisting of 7 sub-specimens in series.

While the strain rate model accurately describes the behaviour of clays when strain is increasing, itis not the case when the axial strain remains constant, as in relaxation (constant strain) tests. Yoshikuniet al. (1994 & 1995) performed special oedometer tests with different phases of consolidation (opendrainage) and relaxation (closed drainage). In one series of three tests, these authors stopped secondaryconsolidation under an effective stress of 314 kPa by closing the drainage and monitored the excess porepressure developing with time. As shown in Figure 9.8, the higher the strain rate at which secondaryconsolidation is stopped, the higher is the generated excess pore pressure, indicating that, when strain

128

I

90 100 110 120 130 140 150 160 170 180

Effectivestress,

σ ' (kPa)

4 3 1σ 'vf

F

PEOPSecondaryconsolidation(C /C = 0.03orm' = 33)

α e c e(m

in-1

)

10

10

10

10

10

-5

-6

-7

-8

-910

-10

1

2

3

4

500

mm

1.90

1.85

1.80

1.75

1.70

1.65

1.60

1.55

Voi

dra

tio,

e

2Primary consolidationin the sub-elements

e -σ '

3 x 10103 x 1010

v

-6

-6-7

-7

e = 10-5 min-1

min

Figure 9.7. Consolidation of the Saint-Hilaire clay for pressure increment from 97 kPa to 138 kPa (after Mesriet al., 1995; from Leroueil & Marques, 1996).

150

100

50

00 5 10 15 20 25

Time t (10 min.)

Exc

ess

po

rep

ress

ure

u(k

Pa

)

Case 1

Case 2

Case 3

Vo

idra

tioe

Vertical effective stress(kPa, log scale)

Case 1

Case 2

Case 3

10

10

10

e(min. )

-3

-4

-5

2(a) (b)

-1

Figure 9.8. Excess pore pressure generation during relaxation tests under one-dimensional conditions (fromYoshikuni et al., 1994).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 128

rate decreases, the tendency to creep also decreases. Holzer et al. (1973) report similar results underisotropic stresses. One explanation for the behaviour observed in relaxation tests is that the strain-ratemodel (Equations 9.1 to 9.3) applies only to the plastic component of strain, so that, in relaxation testswhere the total strain � is constant, the increase in plastic strain �p associated with creep is compensatedby a decrease in elastic strain �e, and thus a decrease in effective stress (Figure 9.9).

� � �e � �p � constant (9.6a)

where �e and �p are respectively the elastic and plastic components of strain.So, in relaxation tests:

�·e � ��·p (9.6b)

The rate of change in pore pressure (u· ) following closing of the drainage when the soil is in second-ary consolidation at a strain rate �· � ��·p (point A in Figure 9.9) would be:

(9.7)

in which � is the effective stress at the time of closing drainage and Cs is the recompression index.When the applied load is decreased and soil swells, viscous effects are also observed. After primary

swelling associated with pore pressure equalization, secondary deformations in swelling or in compres-sion develop. Such behaviour has been observed and discussed by many authors (Mesri et al., 1978;Urciuoli, 1992; Feijo & Martins, 1993; Yoshikuni et al., 1993). It can be illustrated by the resultsreported by Feijo & Martins. Special oedometer tests were performed on organic Sarapui clay fromBrazil (see Almeida & Marques, 2002*), in which several specimens were first loaded in the normallyconsolidated range and then, at the end-of-primary consolidation, unloaded in one step to various over-consolidation ratios. The specimens were left under constant stress for more than 5 months and theobservations are shown in Figure 9.10. For the specimens at OCR of 1.5 and 2, after some swelling, thesoil starts to compress again in a manner similar to what is observed with normally consolidated clays.For OCRs of 6, 8 and 12, the soil shows some secondary swelling during the entire test. At an OCR of4, secondary deformations are not significant. This kind of “neutral OCR” may vary from one soil toanother and certainly with the strain rate at the time of unloading.

9.2 Peak strength envelope and limit state curve

Sällfors (1975) and Tavenas & Leroueil (1977) indicated that the effects of time and strain rate on thepreconsolidation pressure can be generalized to the entire limit state curve of the soil. As indicated in

� �ue

Cs

p��

�2 3 1 0. ( ) ′

129

A (ε)du

dε

ε

C1 + e

log σ'1

p

e

v

o

dε

s

Figure 9.9. Strain components in relaxation test.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 129

Figure 9.2, secondary consolidation, which induces a decrease in void ratio and thus an increase in pre-consolidation pressure also induces an increase in size of the entire limit state curve. Also, for an agednatural clay, the slower the stress changes towards the limit state curve, the smaller the stress conditionsshould be at gross yielding.

Triaxial tests performed by Lo & Morin (1972) on St-Vallier clay (Figure 9.11) demonstrate a strainrate effect on the strength envelope of the overconsolidated soil, i.e. the upper part of the limit state

130

-8

-6

-4

-2

0

2

(sw

elli

ng

)(c

om

pre

ssio

n)

Vo

lum

etr

icst

rain

,ε(%

)v

-1 2 3 4 5 610 1 10 10 10 10 10 10

Time (min.)

OCR12.0

8.0

6.0

4.0

2.0

1.5

Figure 9.10. Volumetric strains observed after unloading of Sarapuí clay (after Feijó & Martins, 1993).

100

50

00 50 100 150 200

(σ'

-σ'

)/2

(kP

a)

ar

CIU, 0.5 %/h

CID, 0.1%

/h

CIU, 0.05%

/h

CID, 0.0025%

/h

CSLσ'

=0

3

(σ' + σ' )/2 (kPa)a r

Figure 9.11. Effect of strain rate on the strength of Saint-Vallier clay (after Lo & Morin, 1972).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 130

curve, similar to that on the preconsolidation pressure. Similar results were obtained for other easternCanada clays (Tavenas et al., 1978; Vaid et al., 1979; Leroueil & Tavenas, 1979).

Marchand (1982) reported triaxial tests performed on Mascouche clay, from eastern Canada. Thepeaks obtained in CIU, CAU and CID tests, and the effective stress applied in triaxial drained creep testsin which failure was reached are shown in the insert of Figure 9.12 with their associated strain rate (min-imum strain rate reached before failure in creep tests; see Section 9.6). All the data are relatively well-organized, confirming an effect of strain rate on the peak strength envelope of overconsolidated andsensitive clay. The one-dimensional CRS and creep test results, also presented on the figure, indicate aneffect of strain rate on the preconsolidation pressure similar to that on the strength envelope. It is thusthe entire limit state curve that is strain rate dependent.

The viscous behaviour of Berthierville clay (see Figure 9.4 for its one-dimensional rheological law)has been investigated by Boudali (1995). Continuous and step-loaded triaxial compression tests wereperformed along K � �3/ �1 � constant stress paths. From the test results and for the different stresspaths followed, the following conclusions were drawn:

– Along the different stress paths, yielding at a given strain rate is well defined, and the mean effectivestress at gross yielding varies with the strain rate in the same manner as the preconsolidation pres-sure (Equation 9.3). This implies that the limit state curves obtained for different strain rates have thesame shape, as shown in Figure 9.13a for Berthierville clay.

– The stress-strain curves obtained at different strain rates along a given K � constant stress path canbe normalised with respect to the gross yield stress value corresponding to the considered strain rate.

– The effective stress-strain-strain rate model proposed by Leroueil et al. (1985a) can be extended tothe entire stress diagram. Such a model can be schematized by a limit state surface having differentpeels, each one being associated to a different strain rate (see Figure 10.8).

131

ε

''

''

CAUCIUCADCAUCA-creepFrom one-dimensionalCRS and creep testsCompression tests atconstant ' / ' ratioσ

= 1.4 x 10 s-6 -1ε1

= 1.4 x 10 s-6 -1ε 1

= 2.3 x 10 s-7 -11

= 6.0 x 10 s-9 -1ε1

σa r

150

100

50

00 50 100 150 200 250 300

+σ σ2

a r''

σσ

- 2ar

ε = 10 svs-6 -1

ε = 10 svs-9 -1

K lineo

(kPa)

(kP

a)

140140

23

3

0.2

0.6

0.2

0.3-7

-8

-9

10

10

1010

ε vs(s

)-1

140-6

Numbers close to the pointsare strain rates in 10 s-8 -1

120

100

80

6070 90 110 130 150

+σ σ2

a r

σσ

- 2

''

(kPa)

(kP

a)

0.6

140

CSL

ar

ε

Figure 9.12. Influence of strain rate on the limit state curve of Mascouche clay (after Marchand, 1982, and Leroueil& Marques, 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 131

It is worth mentioning that the model previously described has limitations. In particular, Hight(1983), Hight et al. (1987) and Sheahan et al. (1996) who performed anisotropically consolidatedundrained triaxial tests on reconstituted Lower Cromer till (Ip � 13%) and Boston Blue clay (Ip � 23.7%)respectively did not observe strain rate effects on the strength envelope of the soil in the overconsoli-dated range (Figure 9.14).

9.3 Stress strain behaviour

9.3.1 Small strain moduliThe effect of strain rate on small strain moduli has been the object of numerous studies (Isenhower &Stokoe, 1981; Rampello & Silvestri, 1993; Zavoral & Campanella, 1994; Stokoe et al., 1995; Tatsuoka

132

40

30

20

10

00 10 20 30 40

p' (kPa)

q(k

Pa

)

101010

-7

-8

-1εvs (s )

-6

T = 20 ˚C

K=

0.43

0.50

0.65

0.80

1.00

40

30

20

10

00 10 20 30 40

p' (kPa)q

(kP

a)

K=

0.43

0.50

0.65

0.80

1.00

= 10-7 -1ε

vs (s )

5˚ 20˚C 35˚

TriaxialOedometer

(a) (b)

Type oftest

Figure 9.13. Variation of the limit state curve of Berthierville clay with: (a) strain rate and (b) temperature (afterBoudali, 1995).

( )

OCR = 1

OCR = 2

OCR = 4

OCR = 8

0.4

0.3

0.2

0.1

0

-0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

+σ σa r''

σσ

-a

r''

2 σ'vm

2σ

' vm

0.0510.505.0

49.0

Symbol ε (%/h)

Peak shear stress

Start of shear

/(

)

/

φ' = 32˚

Figure 9.14. Normalized effective stress paths and strengths as obtained on resedimented Boston Blue clay inundrained compression tests performed at different OCRs and strain rates (after Sheahan et al., 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 132

et al., 1995; d’Onofrio, 1996; and others). Some of these results were presented by Tatsuoka et al. (1997, 2000), Lo Presti et al. (1997) and Tatsuoka (2000), and are shown in Figure 4.4. It can be seenthat the effect of strain rate on small strain Young’s modulus is not very important. From published data,it appears that the strain rate effect is: (a) relatively small and less than 6% per tenfold increase in strainrate in cohesive soils; (b) smaller in cohesionless materials than in clayey soils. Lo Presti et al. (1997),Santucci de Magistris et al. (1999) and Tatsuoka et al. (2001) show that the effect is increasing withstrain. As shown in Figure 9.15 for the Metrano silty sand (Santucci de Magistris et al., 1999), almostinsignificant at a strain of 10�6, the effect of strain rate on the secant Young’s modulus increases withstrain to reach about 5% per logarithm cycle of strain rate at a strain of 1.5 � 10�5.

As indicated in Section 4.4, the increase in Go with time during secondary consolidation is oftencharacterized by NG (Equation 4.12). This parameter is soil dependent. Afifi & Richart (1973) indicatedan increase of NG when the size of the particles, characterized by D50, decreases; Kokusho et al. (1982)indicated a relationship between NG and Ip; Mesri (1987) proposed an equation based on considerationson secondary consolidation and C�e/Cc; Lo Presti et al. (1996) showed a relationship between NG andC�e (Figure 9.16).

133

Figure 9.15. Relationship between the secant Young’s modulus defined at different small strains obtained onMetramo silty sand (after Santucci de Magistis et al., 1999; from Tatsuoka, 2000).

Figure 9.16. NG coefficient versus secondary compression index (from Lo Presti et al., 1996).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 133

9.3.2 Moderate strainsThe influence of strain rate at moderate strains is well illustrated by undrained triaxial test resultsreported by Shibuya et al. (1997) and Shibuya (2001). Figure 9.17a shows the stress-strain curvesdeduced from CIU tests performed on a reconstituted clay (called NSF clay) at strain rates of 0.011,0.133 and 1.40%/min. It can be seen that, at small strains, smaller than about 0.002%, the modulus isabout the same for the three tests; the most rapid test keeps the same modulus up to strain of about0.01% whereas the two other tests show a diversion from the linear behaviour at smaller strains; finally,at strains larger than about 0.01%, the influence of strain rate on stress-strain behaviour is quite clear.Figure 9.17b shows the relationships between Young’s modulus and strain. It also demonstrates thestrain rate dependency with, in particular, a threshold strain at which the stress-strain relationship stopsbeing linear (Y1, see Section 4 and Figure 4.1) which decreases with strain rate. From such observa-tions, Tatsuoka and co-workers developed the model schematized in Figure 9.18. It implies that Y1 andY2 surfaces (Figure 4.1) increase in size with increasing strain rate (Tatsuoka et al., 1997), in fact as thelimit state curve (Y3) does.

Vaid & Campanella (1977) performed a variety of triaxial tests on the undisturbed Haney clay. Inparticular, they performed undrained compression tests at different strain rates and undrained compres-sion tests in which the strain rate was changed at a given strain. As shown in Figure 9.19a, when the

134

Figure 9.17. Results of undrained triaxial tests on NSF clay: (a) stress-strain relationship at small strains; (b) Young’s moduli with stress (after Shibuya, 2001).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 134

strain rate was increased from 3.5 � 10�3 to 4.7 � 10�2%/min, the stress-strain curve moves from thecurve corresponding to the former strain rate to that corresponding to the new strain rate. The transitioncorresponds to a high-stiffness zone. Based on such results in which the strain rate could increase ordecrease, and creep tests, Vaid & Campanella (1977) proposed the following model in which the devi-atoric stress is a function of strain and strain rate:

q � f(�s, �·s) (9.8)

Tatsuoka et al. (2000) show many other examples of shear tests performed on soft and stiff clays,silty sand, sand, mudstone and gravel that generally confirm Equation 9.8. This latter equation thus

135

Figure 9.18. Framework of stress-strain relationship at small strains (Tatsuoka & Shibuya, 1992; Tatsuoka et al.,2000).

Figure 9.19. Undrained triaxial compression tests on undisturbed Haney clay consolidated to �c � 505 kPa;

(a) influence of strain rate; (b) influence of undrained creep (after Vaid & Campanella, 1997).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 135

appears very general. Also clear from these results is the fact that when shear is started after some creepor when the strain rate is suddenly increased, the soil initially shows high stiffness.

The behaviour described in Section 9.1 for oedometer tests refering to axial effective stress, strainand strain rate can certainly be extended to mean stress, volumetric strain and volumetric strain rate asfollows:

p� � f�(�vol, �·vol) (9.9)

Equations 9.8 and 9.9 are very similar and probably are two facets of the same viscous phenomenon.They are schematized in Figures 9.20a and b respectively.

In some cases, the stress-strain curve obtained after a creep phase or when increasing the strain ratemay overshoot the reference curve corresponding to the considered strain rate, as indicated by thedashed lines shown in Figures 9.20a and b. Figure 9.19b shows such an example where the stress-straincurve obtained after undrained creep clearly overshoots the reference stress-strain curve. These phenomena are associated with structuring phenomena and are described in Section 11.

9.4 Critical state line

Tests performed at different rates of strain on normally consolidated samples of clay do not show anysignificant effect of strain rate on the normally consolidated friction angle when determined at( �1/ �3)max (Bjerrum et al., 1958; Vaid & Campanella, 1977; Badra-Blanchet, 1981; Lefebvre &Leboeuf, 1987; Sheahan et al., 1996). Tests performed on the Erksak sand by Been et al. (1991) indicateno significant effect of strain rate on the critical state strength either. The critical state line thus does notseem to be significantly influenced by strain rate in the q–p� diagram.

9.5 Creep

On the basis of creep tests performed in the triaxial apparatus in drained and undrained conditions,Singh & Mitchell (1968) showed that: (a) the logarithm of strain rate decreases with the logarithm oftime; (b) the slope of the log �·a–log t relationship is essentially independent of the applied stress; (c) an increase in applied deviatoric stress shifts the relationship to higher strain rates; and (d) except atvery low and very high stress levels, there is a linear relationship between the logarithm of the strain rateand the deviatoric stress at a given time. Singh & Mitchell (1968) proposed the following equation:

(9.10)

where �a is the axial strain rate at any time t, q� is the stress level equal to the applied deviatoric stressdivided by the deviatoric stress at failure, m is the slope of the log �·a–log t curve which generally falls

�� � �a

q 1m

A et

t⋅ ⋅

136

ε

q

s vol

εs

0.1

0.01 εs

εs

Creep

ε

ε 0.10.01 ε

ε

Creep

σ'

orp

'v

(a) (b)

A B

CD

vol

vol

vol

Figure 9.20. Influence of strain rate on the stress-strain behaviour of soilds: (a) shear behaviour; (b) compressionbehaviour.

09031-02[1].qxd 18/Oct/02 12:10 AM Page 136

between 0.7 and 1.0, t1 is a reference time generally taken equal to 1 min, and � and A are creep param-eters. Murayama et al. (1984) found a similar behaviour for Toyoura sand, with a m value of 1.0.

For the creep tests performed under undrained conditions at high stress levels, the strain rate reachesa minimum value after some time and then increases up to failure of the soil specimen. Vaid &Campanella (1977) showed on Haney clay that the relationship between this minimum strain rate andthe applied deviatoric stress is the same as that obtained in undrained shear between the strain rate usedand the deviatoric stress at the peak.

The studies on creep previously mentioned were mostly performed on normally consolidated materials. A few studies were performed on overconsolidated clays, thus inside the limit state curve(Bishop & Lovenbury, 1969; Larsson, 1977; Tavenas et al., 1978; d’Elia, 1991). Typical creep testresults obtained by Tavenas et al. (1978) on Saint-Alban clay are shown in Figure 9.21. They show a lin-ear decrease of the logarithm of strain rate with logarithm of time and thus extend the validity of

137

A B

C

D

E

F

G

10

10

10

10

10

101 10 10 10 10 10

5432

-6

-5

-4

-3

-2

-1

Time t (min.)

Axi

als

tra

inra

teε

(%/m

in.)

a

σ' = 25.0 ;σ' = 16.5ABCDEFG

σ' = 34.0 ;σ' = 16.5

σ' = 41.0 ;σ' = 16.5σ' = 52.8 ;σ' = 16.5σ' = 33.1 ;σ' = 10.3σ' = 33.9 ;σ' = 3.1σ' = 25.2 ;σ' = 2.0

1 3

1 31 3

1 3

1 3

1 3

1 3

AB

CE

D

FG

Limit state curve

5

4

2

1

0

4-5 3

2-3

0 10 20 30 40 50-10

0

10

20

(σ' + σ' )/2 (kPa)a r(σ

'-σ

')/

2(k

Pa

)a

r

Saint-Alban, 3 m

Figure 9.21. Axial strain rate-time relationship for creep tests on the Saint-Alban clay (after Tavenas et al., 1978).

09031-02[1].qxd 18/Oct/02 12:10 AM Page 137

Equation 9.10 to the overconsolidated domain. Tavenas et al. (1978) considered, however, that the stressfunction (exp(�q�)) used in Equation 9.10 should be generalized and written as n( �), with n( �) beinga stress function reflecting the shape of the limit state curve of the soil. Equation 9.10 then becomes:

(9.11)

Equation 9.11 would be valid for both volumetric and shear strains. For high stress levels, Equations9.10 and 9.11 are no longer valid after some time. As shown on Figure 9.21 for tests D, F, and G, the rateof strain reaches a minimum value and then increases to reach failure. From the data presented byLefebvre (1981), the minimum strain rate is reached at a strain approximately equal to that obtained atfailure in conventional triaxial drained tests, indicating that accumulated strain is a major factor in thecontrol of failure.

The response of soils in terms of variation with time of volumetric strain in drained tests or pore pres-sure in undrained tests depends on the overconsolidation ratio of the soil. For example, in Figure 9.21,the void ratio decreased with time for test C, and increased with time for test G.

As described in this section, the viscous behaviour of soils would be time dependent whereas previ-ous results indicated that soil behaviour is strain rate dependent. In order to clarify this aspect, Leroueil(1998, 2001) established a simple model and simulated creep tests. The hypotheses considered were: a hyperbolic stress-strain curve; a log-log variation of shear strength with time; and failure controlledby a critical accumulated strain, 0.9% in that case. The model, with the considered equations and param-eters, is shown in Figure 9.22a. Creep tests have been simulated for the deviatoric stresses indicated byarrows in Figure 9.22a, and the results are shown in Figure 9.22b. It can be seen that the plots of loga-rithm of strain rate against logarithm of time lines are linear, as observed in laboratory tests (Figure 9.21),and as indicated by Equations 9.10 and 9.11. As indicated by Leroueil (2001), this means that, funda-mentally, soil behaviour is strain rate dependent and that the time model proposed by Singh & Mitchell(1968) reflects both this strain rate dependency and the test itself, in which stress conditions for creeptests are suddenly applied at time zero. It is worth noting that the strain rate model shows a slightdecrease of the slope, m, of the log �� against log t curves, as observed in laboratory creep tests.

Behaviour similar to that described for soft clays was also observed for stiff clays (D’Elia et al.,1998) and soft rocks (Nishi et al., 1983; Adachi et al., 1998).

9.6 Some other implications of the viscous behaviour of soils

The viscous behaviour previously described has many possible consequences on soil behaviour. Someare described in the following paragraphs.

�� � A nt

t1

m

⋅ ⋅

( )

138

Figure 9.22. Simulation of creep tests: (a) strain rate dependency of the stress-strain behaviour and input parame-ters; (b) simulated creep tests for the conditions given in (a) (from Leroueil, 1998, 2001).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 138

9.6.1 Undrained creep under constant deviatoric stressUndrained creep of normally consolidated soils, under deviatoric stress or under isotropic stress condi-tions, is accompanied by an increase in pore pressure and thus by a decrease in effective stress(Arulanandan et al., 1971; Holzer et al., 1973). Arulanandan et al. (1971) carried out undrained creeptests at different stress levels on San Francisco Bay mud. The effective stress conditions at differenttimes are shown in Figure 9.23. It can be seen that the locus of effective stress conditions at a given timemaintains essentially the same shape, but progressively move towards smaller stresses as time increases.This behaviour can be explained in the same manner as for relaxation tests in one-dimensional condi-tions (see Figure 9.8 and Equations 9.6 and 9.7).

As indicated, in particular by Sekiguchi (1977) and Sheahan (1995), there must be a lower limit tostress conditions which can be reached during creep tests. This is supported by observations showingthat highly overconsolidated clays have a tendency to generate negative pore pressures in undrainedcreep tests, or swelling in drained creep tests (see Figure 9.10). Sheahan (1995) calls the lower limit“static yield surface”; “inviscid yield surface” has also been used in the literature.

9.6.2 Undrained relaxation (constant �a) in triaxial conditionsThese tests are characterized by the fact that the strain, the axial strain in triaxial tests, remains constantwith time. Such tests have been performed on clays and sands by Lacerda & Houston (1973), Akai et al.(1975), Lacerda (1976), Hicher (1988) and Sheahan et al. (1994). The results show a linear decrease ofthe deviatoric stress with the logarithm of time; they also show that pore pressure does not vary signif-icantly during relaxation tests. This implies, as demonstrated in Figure 9.24, that the effective radialstress remains approximately constant.

9.6.3 Undrained triaxial tests performed at different strain ratesIf undrained triaxial tests are performed on normally consolidated clays at different strain rates, the gen-erated pore pressure increases as the strain rate decreases. As a result, a rapid test will reach failure at apoint such as Rnc in Figure 9.25 whereas a slow test will reach failure at Snc. As the critical state strengthenvelope is strain rate independent (see Section 9.4), Rnc and Snc are on the same strength envelope andthe difference in strength is due to different pore pressures at failure (Figure 9.25). For undrained triax-ial tests performed in the overconsolidated domain, the strength results from the pore pressure gener-ated during creep and a strength envelope that may be strain rate dependent (Figure 9.25). These effectsresult in a decrease in undrained shear strength of natural clays with decreasing strain rates. Compilingtest results obtained on 26 different clays, Kulhawy & Mayne (1990) found a typical change inundrained shear strength of 10% per logarithm cycle of strain rate (Figure 9.26). Graham et al. (1983a)found similar variations in triaxial compression, but also in triaxial extension and direct simple sheartests. However, studies performed by Hight (1983), Hight et al. (1987) and Sheahan et al. (1996) onreconstituted Lower Cromer till and Boston Blue clay indicate that the effect of strain rate on the com-pression undrained shear strength considerably decreases when the overconsolidation ratio increases.

139

Figure 9.23. Change in effective stress with time during undrained triaxial creep tests (( 1 � 3) � cst) (afterArulanandan et al., 1971).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 139

140

+σ σ1 3'' 2 σ'vm

0 0.2 0.4 0.6 0.8 1.0 1.20

0.2

0.4

0.6

SR1-8SR1-9SR1-5

( ) /

σσ

13'

'2

σ'vm

()

/

Figure 9.24. Effective stress paths followed during triaxial relaxation tests (�1 � cst) on normally consolidated SanFrancisco Bay Mud (data from Lacerda, 1976; after Martins, 1992).

I

I

RR

Y

Y

SS

RapidSlow

Undrained tests

Y: limit state curve atconstant void ratio

nc

nc

oc

ocR

S

oc

nc

f ' line

nc

( )s s+ /2''a r

()

ss

-/2

'' a

r

Figure 9.25. Undrained compression tests – Schematic behaviour.

Figure 9.26. Influence of strain rate on the undrained shear strength measured in triaxial compression (fromKulhawy & Mayne, 1990).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 140

9.6.4 Secondary consolidation and subsequent undrained compression testsThe influence of viscous effects on consolidation and subsequent shear testing is shown in Figure 9.27for soil specimens consolidated at point A, in the normally consolidated domain (Figure 9.27a), andsubjected to undrained compression. If the specimen is tested immediately after reaching the end-of-primary consolidation, the stress path is as AD in Figure 9.27a and corresponds to A�D� in Figure 9.27b,with D� on the critical state line of the soil. If, for a second specimen, the drainage valve is closed at theend of primary consolidation and the soil left under constant total stresses during some time, pore pres-sure develops (there is creep as shown in Figure 9.23) and the effective stresses move from A to B.During shearing, the soil specimen first exhibits an “overconsolidated behaviour” with a stress path ini-tially at constant p� (assuming a perfectly linear elastic behaviour), and then moves towards D. Since thevoid ratio is the same as for the first test, the shear strength is the same for both specimens. If, for a thirdspecimen, drainage is allowed for some time after the end-of-primary consolidation, the void ratiodecreases and soil conditions progressively move from A� to C� in Figure 9.27b. The new limit statecurve for the soil is then RN (Figure 9.27a) with its projection in the e–p� diagram being R�N�. At pointC�, the soil is overconsolidated and behaves as such during the initial part of the undrained shear test(constant p�); then, it progressively goes to failure, at point E (Figure 9.27a) and E� (Figure 9.27b) cor-responding to its void ratio. This was demonstrated by Shen et al. (1973).

141

Figure 9.27. Effect of time on consolidation and subsequent undrained compression (from Leroueil & Marques,1996).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 141

9.7 K0

As seen in Section 16.7, the coefficient of earth pressure at rest for normally consolidated soils,Konc � �h/ �v, can be described by Equation 16.9 (also see Figure 16.6). One question raised bySchmertmann (1983) was how, if at all, K0 varies with time during secondary consolidation or aging. Infact, experimental data presented in the literature clearly show that K0 increases with time in normallyor slightly overconsolidated clays (Tavenas & Leroueil, 1987). Mesri & Hayat (1993) compiled �K0/�log tvalues measured during secondary consolidation and found that they range between 0.003 and 0.05.

9.8 Residual strength

The residual strength envelope is characterized by a friction angle ��r and a cohesion equal to zero(Section 8.4). Petley (1966), Kenney (1967), Lupini (1980) and Lemos (1986) studied the effect of therate of shearing on the residual friction angle of clays. The effect of rate of displacement is small, gen-erally smaller than 3% per logarithm cycle for rates smaller than 10�1mm/min. At faster rates (largerthan 1–10 mm/min), the residual strength under a constant normal stress has been observed to remainconstant or to increase or to decrease, depending on the soil and testing conditions. Tika et al. (1996)studied these phenomena, and concluded that decrease in residual strength at fast rates would be due toan increase in void ratio in the shear zone, and would exist if free water is available. For sands, Hungr &Morgenstern (1984) did not find any rate effect.

10 INFLUENCE OF TEMPERATURE

Temperature has two major effects on soils: thermal expansion of solid particles and pore water, andthermally induced modification of the strength of contacts between particles or aggregates. The com-bined effect of these phenomena on the temperature-volume-effective stress behaviour of soils in bothdrained and undrained conditions is explained by Mitchell (1993) and will not be described here indetail. Figure 10.1, deduced from tests on illite samples initially isotropically consolidated under200 kPa, illustrates the behaviour generally observed. In drained conditions, an increase in temperatureis associated with an expulsion of pore water from the clay sample whereas a decrease in temperature isassociated with absorption. As explained by Mitchell (1993), “the weakening effect of the higher tem-perature is compensated by the strengthening effect of the lower void ratio”. However, due to changesin the particle arrangement, the process is not reversible for soils normally or slightly overconsolidated.In undrained conditions, heating is associated with an increase in pore pressure, and consequently, with

142

46 32

59 46 31 19

32

46

60

32

100 125 150 175 200 225

0

0.5

1.0

1.5

2.0

Effective stress (kPa)

19

46

19

Undrained

dra

ine

d

19: temperature in ˚C

I

Po

rew

ate

rd

rain

ed

(Pe

rce

nt

of

initi

als

am

ple

volu

me

)

Figure 10.1. Effects of temperature changes on saturated illite under drained and undrained conditions (initialisotropic effective stress � 200 kPa) (after Mitchell, 1993).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 142

a decrease in effective stress. In these conditions where there is no significant particle rearrangement,the process is almost reversible.

The results presented and discussed in this Section are associated with temperatures from above zeroto 100°C, i.e. temperatures that are not low enough to modify soil fabric by the effect of frost and nothigh enough to produce changes in mineralogy.

10.1 One-dimensional compression

In recent years, many studies have been conducted on the effects of temperature on the compressibilityof natural soft clays (Eriksson, 1989; Tidfors & Sällfors, 1989; Boudali et al., 1994; Marques, 1996;Graham et al., 2001; Marques et al., 2002). Typical CRS test results, obtained on sulphide clay fromLulea, Sweden, are shown in Figure 10.2. It can be seen that there is a significant effect of temperatureon the compressibility of the clay. With increasing temperature, the soil becomes more compressible inthe overconsolidated range, the preconsolidation pressure decreases, and the entire compression curvemoves towards smaller effective stresses. It can be seen on Figure 10.2b that the effect is more impor-tant at temperatures below 35°C than at higher temperatures. The two CRS tests A and B performed onthe St-Polycarpe clay (Figure 9.3b) with changes in temperature at different strains from 5°C to 20°C in Test A and from 50°C to 20°C in Test B, show the effective stress-strain curves jumping from one

143

10 100Vertical effective stress (kPa)

0

5

10

15

20

25

30

0 10 20 30 40 50 60Temperature (°C)

80

70

60

50

40

30

20

Pre

con

solid

atio

np

ress

ure

(kP

a)

Axi

als

tra

in(%

)

(a)

(b)

45 35 25

T(°C

)515

Figure 10.2. Oedometer tests performed on Luleå clay at various temperatures (from Eriksson, 1989).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 143

constant temperature curve to another, and thus confirm the influence of temperature on one-dimensional compression.

Boudali et al. (1994) performed CRS tests at different strain rates and temperatures on 3 clays,among which was the Berthierville clay previously discussed (Figure 9.4). Their results show the fol-lowing: (a) the preconsolidation pressure can be described as a function of strain rate and temperature(Figure 9.4a); and (b) the effective stress-strain curves obtained at various strain rates and temperaturesreduce to a unique one when normalized with respect to the preconsolidation pressure corresponding tothe strain rate and temperature used in each test (Figure 9.4b). The model proposed by Leroueil et al.(1985a) for strain rate effects can thus be extended for including temperature effects. Equations 9.1 and9.2 then become:

�p � f(�1, T) (10.1)

and

�1 � �p(��1, T) � g(�1) (10.2)

The test results shown in Figures 9.3 and 10.2 fit this model, which also implies that the compressionindex (or ) is independent of strain rate and temperature.

Figure 10.3 shows the preconsolidation pressure (or the effective stress at a given void ratio) nor-malized with respect to the preconsolidation pressure (or the effective stress at the same void ratio)measured at 20°C, as a function of temperature. The change in preconsolidation pressure with temper-ature is about the same for all the clays considered, indicating a change in preconsolidation pressure ofalmost 1% per °C between 5°C and 40°C, and much smaller at larger temperatures.

Equations 10.1 and 10.2 indicate that strain rate and temperature are two factors which are intimatelylinked in the viscous deformation of soils and influence the maximum effective stress a soil can supportat a given void ratio. This results from fundamental physics and can be explained, at least qualitatively,by the rate-process theory (Murayama & Shibata, 1961; Mitchell, 1964, 1993).

10.2 Limit state curve

To investigate the effect of temperature on the limit state curve, Boudali (1995) performed, in additionto the oedometer tests previously described (Figure 9.4), isotropic compression tests and undrained tri-axial tests at temperatures of 5, 20 and 35°C on Berthierville clay. The isotropic compression tests showa behaviour very similar to that observed in oedometer tests with an isotropic yield stress decreasingwith increasing temperature (Figure 10.4). Burghignoli & Desideri (1988) on reconstituted Todi clay,

144

1.40

1.20

1.00

0.80

0.600 20 40 60 80 100

Temperature (°C)

Boudali et al., 1994Despax, 1975Campanella & Mitchell, 1968Tidfors & Sällfors, 1989Eriksson, 1989Moritz, 1995a and bMarques, 1996Akagi & Komiya, 1995

σ'/σ

'(T

=2

0˚C

)p

pσ'

/σ'

(T=

20

°C)

11

or

Figure 10.3. Variation of the normalized preconsolidation pressure, or vertical effective stress at a given void ratio,with temperature (from Leroueil & Marques, 1996).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 144

Graham et al. (2001) on illite and Marques et al. (2002) on St-Roch-de-l’Achigan clay observed simi-lar effects of temperature. The isotropic yield stresses and preconsolidation pressures obtained onBerthierville clay at a strain rate of 10�7 s�1 are presented in Figure 9.13b (the preconsolidation pres-sures deduced from Figure 9.4 have been plotted, in an approximative manner, on the K � 0.5 line).Even if the data available are limited, it seems that the effect of temperature on the limit state curve isvery similar to the effect of strain rate (Figure 9.13a).

Due to such effects of temperature, for a soil consolidated in the normally consolidated range at agiven temperature, cooling causes the soil to behave as if it were overconsolidated (Plum & Esrig, 1969;Burghignoli & Desideri, 1988; Hueckel & Baldi, 1990). As a result, volumetric strains due to tempera-ture changes in the overconsolidated range are relatively small and to a large extent reversible (Hueckel &Baldi, 1990; Burghignoli et al., 1992; Boudali et al., 1994). Plum & Esrig (1969), Hueckel & Baldi(1990) and Burghignoli et al. (1992) also showed that these volume changes depend on the overconsol-idation ratio. For the natural Pasquasia clay (Figure 10.5) and for temperatures increasing from 21°C to

145

0 20 40 60 80

σ' (kPa)

0

2

4

6

8

10

12

14

16

Vo

lum

etr

icst

rain

(%)

T (˚C)

5202035

Figure 10.4. Effective stress-volumetric strain relations at a strain rate of 10�7 s�1 and at different temperatures,isotropic compression isotropic compression of Berthierville clay (from Boudali, 1995).

100

80

60

40

20

Tem

pe

ratu

re(˚

C)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3Volumetric strain (%)

13.07 5.67 3.4OCR = 7.5

Figure 10.5. Drained heating tests at constant effective stress at various OCRs (in terms of isotropic stress). NaturalPasquasia clay (from Hueckel & Baldi, 1990).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 145

more than 90°C, volumetric strains are positive for OCRs smaller than approximately 6, and negativefor higher OCRs. The similarity with the tendency for the soil to creep in compression when slightlyoverconsolidated and in swelling when strongly overconsolidated (see Figure 9.10) is noteworthy. So, asFeijo & Martins (1993) suggested for creep (see Section 9.1), a “neutral OCR” could also be thought offor temperature.

Hueckel & Baldi (1990), Boudali (1995), Moritz (1995) and Marques et al. (2002) observed a gen-eral tendency for the peak shear strength to decrease when temperature increases. However, as noted byHueckel & Baldi (1990), when temperature increase is associated with a significant void ratio decrease,the strength gain due to this decrease in void ratio could compensate for the strength loss due to a highertemperature.

Figure 10.6 summarizes test results obtained by Marques et al. (2002) on St-Roch-de-l’Achigan clayfrom Québec. Below the critical state line, the effect of temperature is very clear. As for peak strengthenvelopes, that at 10°C is the highest; those at 20°C and 50°C do not show a clear effect of temperaturebut that could be due, as suggested by Hueckel & Baldi (1990), to a combined result of weakening ofthe interparticle contacts and strengthening of the soil due to a decrease in void ratio when the soil isheated from 20°C to 50°C.

Graham et al. (2001) performed a variety of triaxial tests on an isotropically consolidated illite attemperatures of 28, 65 and 100°C. Figure 10.7 shows the peak strength envelope obtained in the over-consolidated range, normalized with respect to p�e(T), the mean stress on the normal consolidation linecorresponding to the test temperature, at the considered void ratio. There is some scatter but normal-ization seems to be a reasonable approach.

Equation 10.1 indicates that the preconsolidation pressure is a function of both strain rate and tem-perature. The results presented in Section 9.2 and here show that it can be generalized to the entire limitstate curve and surface. As schematized in Figure 10.8 in a e–p�–q diagram, the limit state surfaceappears to have different peels, each one being associated with different strain rate and temperature. Theresults also indicate that the mechanical behaviour could be normalized with respect to �p or p�e, valuethat would be a function of strain rate and temperature.

10.3 Small strain behaviour

Anderson & Richart (1974) performed low amplitude resonant column tests on cohesive soils at tem-peratures of 22°C and 4°C. Seven soils with plasticity index varying from 12 to 64 were tested; five ofthese soils were undisturbed and two were reconstituted. Anderson & Richart (1974) found shear wavevelocities at 4°C from 0% to 12.5% higher than at 22°C and concluded that temperature has a minorinfluence on shear wave velocity. This is consistent with the very small effect of strain rate on smallstrain shear modulus (see Section 9.3). However, at larger strains (�0.1%), as for strain rate, there isevidence that moduli decrease when temperature increases (Eriksson, 1989; Hueckel & Baldi, 1990).

146

Figure 10.6. Limit state curves of Saint-Roch-de-l’Achigan clay (4.8–5.8 m) at temperatures of 10, 20 and 50°C(after Marques et al., 2002).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 146

147

Figure 10.7. Normalized peak strength envelope for overconsolidated specimens of illite tested at different tem-peratures (modified after Graham et al., 2001).

Figure 10.8. Influence of strain rate and temperature on the limit state surface of soils.

10.4 Critical state line

Test results presented by Mitchell (1964), Hueckel & Pellegrini (1989), Hueckel & Baldi (1990),Graham et al. (2001) and Marques et al. (2002) do not indicate a significant effect of temperature on thefriction angle of normally consolidated clays. Houston et al. (1985) observed a slight increase (1°)between 4 and 40°C and a much greater increase at higher temperatures. Testing a sand-bentonite mix-ture, Lingnau et al. (1995) found very similar strength envelopes at 26°C and 65°C, and a strength enve-lope lying well above at 100°C. Hence, it seems that the strength envelope of normally consolidatedclays, thus the critical state line in the stress diagram, is essentially temperature independent, at least fortemperatures lower than about 50°C.

Sherif & Burrous (1969) report test results in which specimens were consolidated at room tempera-ture and then sheared in undrained, unconfined compression tests at the same or higher temperature.The results presented in Figure 10.9 demonstrate the influence of water content or void ratio; they also

09031-02[1].qxd 18/Oct/02 12:11 AM Page 147

confirm that, at a given water content, the higher the temperature, the smaller the undrained shearstrength. As the strength envelope does not change significantly at the considered temperatures, theresults also indicate that, at a given water content, the p� value at the critical state decreases when tem-perature increases. The data presented by Lingnau et al. (1995), Graham et al. (2001) and Marques et al.(2002) confirm that point. The general behaviour of soils subjected to different temperatures can thusbe described, as shown in Figure 10.10 with: (a) a unique critical state line in the q–p� diagram (Figure10.10a); and (b) normal compression and critical state lines in an e (or w)–log p� diagram that are tem-perature-dependent.

Figure 10.10 schematically shows the behaviour of clay specimens that would be normally consoli-dated and then subjected to undrained compression at different temperatures. The specimens are firstconsolidated to point A, at temperature T1. At the end of primary consolidation, they are at point A�, onthe normally consolidated line NCL1 corresponding to T1 (Figure 10.10b). A specimen sheared inundrained conditions at this temperature reaches failure at point D (Figure 10.10a) corresponding topoint D� on the critical state line CSL1 corresponding to T1 (Figure 10.10b). If, for a second specimen,the valve is closed and the temperature increased to T2, the pore pressure increases and, due to the differ-ence in thermal expansion of water and soil particles, the void ratio increases slightly. The soil conditionsthen move to point B� on the normally consolidated line NCL2 corresponding to the new temperature. If the soil is sheared at T2, the soil conditions progressively move to point E�, on the critical state lineCSL2 corresponding to temperature T2 (Figure 10.10b). In the stress diagram (Figure 10.10a), the corre-sponding point E is on the same strength envelope as the first specimen since temperature has no influ-ence on the friction angle of the normally consolidated soil, but at a lower undrained shear strength. Fora third specimen heated in drained conditions to T2, there is a decrease in void ratio from A� to C� onNCL2 (Figure 10.10b). During undrained compression at T2, soil conditions move to F� on CSL2.

10.5 Residual strength

Butcher (1975) observed on two different clays that the residual friction angle was not influenced bytemperature changes between 10 and 60°C. None or a small effect of temperature on ��r is consistentwith the observations made for the strength envelope of the normally consolidated soil (Section 10.4)and the fact that ��r is only slightly influenced by strain rate (Section 9.8).

148

Triaxial consolidationcurve

44

42

40

38

36

34

328 10 20 40 60 80 100

Compression strength (psi)

Mo

istu

reco

nte

nt

(%)

150˚F75˚F

100˚F

125˚F

Figure 10.9. Effect of temperature on the undrained strength of kaolinite in unconfined compression (from Sherif& Burrous, 1969).

09031-02[1].qxd 18/Oct/02 12:11 AM Page 148

11 INFLUENCE OF MICROSTRUCTURE

11.1 Microstructure of natural soils and soft rocks

As indicated by many authors, in particular by Burland (1990), Leroueil & Vaughan (1990),Schmertmann (1991), Clayton & Serratrice (1993) and Kavvadas (2000) in their synthesis papers, mostnatural soils and soft rocks are microstructured (bonded), i.e. at a given void ratio, they can sustainstresses higher than could the same material non-microstructured. This is evident for soft rocks; it isnow recognized for most soft clays (Tavenas & Leroueil, 1985; Burland, 1990); there is also evidenceof microstructure in sands (Mitchell & Solymar, 1984; Schmertmann, 1991) as well as in stiff overcon-solidated clays and clay shales (Calabresi & Scarpelli, 1985; Picarelli et al., 2002*). Leroueil &Vaughan (1990) concluded that microstructure is as important as initial void ratio and stress history indetermining the behaviour of natural geomaterials. This has been confirmed since that time and specificaspects of microstructured soils have been studied in detail.

There are several causes for the development of microstructure in soils and soft rocks: secondarycompression, thixotropy, cementation, cold welding, etc. As indicated in Section 9, secondary com-pression only gives a strength increase related to the decrease in void ratio. The other factors give thesoil a strength which cannot be accounted for by void ratio and stress history alone.

149

e

D'E'

NCL

A'B'

CSL

C'

P'

P'AC

B

D

E

CSLq

(a)

(b)

NCL

CSL

12

1

2

F'

1 for T2 for Twith T < T1 2

12

FStress pathsin CIU tests

Figure 10.10. Effect of temperature change on consolidation and subsequent undrained compression.

09031-02[2].qxd 18/Oct/02 12:12 AM Page 149

Figure 11.1 summarizes the main differences in mechanical behaviour between microstructured soilsand “ideal soils” as described by CSSM (Section 2). Compared to the same soil non-microstructuredand at the same void ratio, the microstructured soil has higher preconsolidation pressure, undrainedshear strength and stiffness. These features are detailed in Section 11.2; Section 11.3 describes whathappens if the bonds between particles or aggregates are broken and the soil is “destructured”

11.2 Main features of microstructured soils and soft rocks

11.2.1 One-dimensional compressionIn one-dimensional compression, microstructure can be shown by comparison of the compressioncurves obtained on the natural, undisturbed, soil or soft rock and on the same geomaterial, remouldedand reconstituted. This can be illustrated by test results obtained on Mexico City and the GrandeBaleine soft clays (Figure 11.2), hard Pappadai clay (Figure 11.3) and calcarenite (Figure 3.11a).Because of microstructure, these soils can reach a domain of the e–log �v space that is not permittedfor the non-microstructured soil; the apparent preconsolidation pressure of the soil is higher than thevalue normally associated with its void ratio. For the case of Pappadai clay (Cottechia & Chandler,1997; Cotecchia, 2002*), the vertical effective stress at the depth of sampling was 415 kPa while thegeological preconsolidation stress was about 1300 kPa (Point P on Figure 11.3). It can be seen that P lieson the right of the compression curve of the reconstituted clay, indicating that the soil was already struc-tured at the time of the formation of the deposit. The vertical effective gross yield stress deduced from

150

φ' line

C ~ plastic~ elastic

FailureC

~ elastic

Failure~ plastic

e eLog p' Log p'

Yield Yield

~ elastic ~ elastic

~plastic

~p

lastic

Ideal soil Structured soil

(a) (b)

(c) (d)

RL

R LY

Limit stateor yield curve

Limit stateor yield curve

σ' + σ'

2a r σ' + σ '

2a r

σ'-

σ' 2a

r

φ' line

σ'-

σ' 2a

r

Figure 11.1. Schematic behaviour of ideal and structured soils (from Leroueil, 1992).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 150

oedometer tests is about 2600 kPa, twice the geological preconsolidation pressure, confirming thestrong microstructure of Pappadai clay. In dense materials, however, the influence of microstructure ongross yielding may be difficult to observe as indicated by Aversa et al. (1993) and Huang & Airey(1993).

Burland (1990) showed that the void ratio-effective overburden stress relationship for normally con-solidated deposits (sedimentation compression line) is different from the compression line obtained inthe laboratory with reconstituted clay (intrinsic compression line, ICL; see Section 16.3). As shown inFigure 18.2, under a given effective stress, the natural void ratio is higher than that of the reconstitutedmaterial.

Also associated with microstructure is the fact that the compression index Cc is generally larger thanthat of the same soil reconstituted. This aspect is discussed in Section 18.4.

151

Figure 11.2. One dimensional compression curves.

1.3

1.1

0.9

0.7

0.5

0.3

Vo

idra

tioe

10 10 10 10

Reconstituted clay

PPreconsolidation

Yield

Naturalclay

2 3 4

σ ' (kPa)v

Figure 11.3. One-dimensional compression behaviour of the natural and reconstituted Pappadai clay (afterCotecchia & Chandler, 1997; Cotecchia, 2002*).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 151

11.2.2 Limit state curve and peak strength envelopeThe peak strength envelope, and in particular the effective cohesion, of soils in their overconsolidateddomain reflects their microstructure. Figure 11.4 shows the strengths envelopes of stiff Vallerica clay, asdeduced from two series of tests performed by Amorosi & Rampello (1998). In both series, the clay wasfirst anisotropically consolidated to a given vertical effective stress and then unloaded in approximatelyKo conditions before shearing in drained or undrained conditions. In the first series (MP tests), the max-imum vertical stress reached during the tests was smaller than the vertical gross yield stress evaluatedfrom oedometer tests and the soil remained undisturbed. In the second series (HP tests), the soil wasfirst loaded to stresses well in excess of the vertical gross yield stress, and the soil was then destruc-tured. In Figure 11.4, the strength envelopes are normalized with respect to p�e to take into account theinfluence of void ratio. It can be seen that the strength envelope of the undisturbed material (MP tests)is well above the strength envelope of the destructured soil (HP test).

Burland et al. (1996) compared the strength envelopes of four different soils in intact and reconsti-tuted conditions and clearly showed that, at the same void ratio, the strength envelope of the intact soilis above that of the reconstituted soil (see Section 18.6 and Figure 18.7).

11.2.3 Small strain shear modulusMicrostructure reinforces the links between particles, and so increases the small strain shear modulus,even at the same void ratio. This has been shown for clays (Rampello & Viggiani, 2001), soft rocks(Nishi et al., 1989), sands (Katayatama et al., 1986; Hatanaka & Uchida, 1995) and gravels (Goto et al.,1992). Examples are presented in Figures 11.5 and 11.6. In both cases, the cemented soil has highershear modulus than the uncemented one under the same confining stress. Also, it can be seen in Figure11.5 that, as the cement content increases, the small strain shear modulus increases and the influence ofconfining stress decreases. There are other indications in the literature that, in microstructured soils, Godoes not depend only on stresses as suggested by Equation 4.10 but also on the strength of the bonds(Nishi et al., 1989; Rampello & Silvestri, 1993; Stokoe et al., 1994; Tatsuoka et al., 1997; D’onofrioet al., 1998). Hardin (1978) suggested that for clays, the shear modulus at very small strains, Go,depends on applied stresses, void ratio and overconsolidation ratio (OCR). Viggiani & Atkinson(1995a) showed that Go could be defined as follows for isotropically consolidated reconstituted clays:

(11.1)G

pA

p

p

p

po

r r

np

r

m

� �

�

��

�

152

0.8

0.6

0.4

0.2

00 0.4 0.6 0.8

p'/p'

HP testsMP tests

e

q/p

' e

Figure 11.4. Normalized stress paths and strength envelopes of Vallericca clay undisturbed (MP tests) and destructured (HP tests) (from Amorosi & Rampello, 1998).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 152

153

Figure 11.5. Dependence of Gmax in sand on confining stress and cementing (unknown).

Figure 11.6. Effect of cementation of a sand and of unloading-reloading on shear wave velocity (after Fernandez &Santamarina, 2001).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 153

where p� is the mean effective stress, p�r is a reference pressure, p�p is the stress at the intersection of the swelling line with the isotropic normal consolidation line and A, n and m are non-dimensional soil parameters. The ratio (p�p/p�) is thus an overconsolidation ratio defined in term of isotropic stress, Ro.

If Go (at p�p) is the shear modulus when p� � p�p, then:

(11.2)

From the values obtained on reconstituted clays and reported by Viggiani & Atkinson (1995a), (n � m)would vary between 0.45 and 0.56, with an average value of 0.51. Thus, Equation 11.2 could be approx-imated by (Leroueil, 2000):

(11.3)

Equation 11.3 is represented by the dash line in Figure 11.7.From results presented in the literature and having Equation 11.3 as a reference, Leroueil (2000)

indicated that the influence of microstructure on Go could be summarized as shown in Figure 11.7; thehigher the degree of microstructuration, the less is the influence of applied stresses on Go.

11.2.4 Large deformation strengthA general observation made with microstructured soils is that stress conditions obtained in shear testsat large deformation are not on the limit state curve, as schematized in Figure 11.1a (Point C), butinside, as schematized in Figure 11.1b. This is shown for soft clays in Figures 3.2 and 11.8a, finegrained tuff in Figure 11.8b, hard Pappadai clay in Figure 11.9, and microstructured sands (Cuccovillo& Coop, 1997). It is due to the fact that the limit state curve reflects the strength of the bonds while, atlarge strains, most of these bonds are broken.

G

G (at p )

p

p

1

Ro

o p p o� ��

��

G

G (at p )

p

po

o p p

(n m)

� �

�

��

154

Figure 11.7. Schematic figure showing the influence of microstructure on Go (after Leroueil, 2000).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 154

Smith et al. (1992) demonstrated the same aspect by plotting radial stress paths starting from in situstress conditions in a normalized p�/p�e–q/p�e diagram. As shown in Figure 11.10, the stress paths travel from in situ stress conditions to gross yielding and then move back. Cuccovillo & Coop (1993)observed a similar behaviour on Rankin calcarenite; Callisto & Calabresi (1998) also observed it onPisa clay but to a less extent, Pisa clay being less sensitive than Bothkennar clay; in the relatively insen-sitive Queenborugh clay phnomenon is not observed at all (Jardine et al., 2002*).

11.3 Destructuration

By definition, destructuration corresponds to breaking down of the bonds between particles or aggre-gates and thus to a general decrease in strength of the material. Even if destructuration is a progressiveprocess, it mostly occurs when a stress path reaches the limit state (or gross yield, Y3 in Figure 4.1)

155

y

Yield curve

φ' line

120

80

40

00 40 80 120

a) Saint-Vallier clay (after Lefebvre, 1970)

(σ'

–σ'

)/2

(kP

a)

ar

(σ' + σ' )/2 (kPa)a r

CIUCID

Isotropicconsolidation

PeakLargestrain

Yield

100

60

20

20 60 100 140 180160 200

φ' line

Yield curve

Large strainenvelope

Large strainenvelope

CIUCID

Isotropicconsolidation

PeakLargestrain

Yield

(σ'

–σ'

)/2

(MP

a)

ar

12

10

8

6

4

2

00 2 4 6 8 10 12 14 16 18 20 22 24

b) Fine grained tuff (after Aversa et al., 1991)

(σ' + σ' )/2 (MPa)a r

Figure 11.8. Effective stress paths in undrained traxial tests.

09031-02[2].qxd 18/Oct/02 12:12 AM Page 155

156

CSLNormalizedlimit statecurve

Undrained shearingDrained shearingIso. and aniso. compression

2.8

2.4

2.0

1.6

1.2

0.8

0.4

00 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

p'/p'

q/p

'

e

e

Figure 11.9. Normalized stress paths of the natural Pappadai clay consolidated to states before isotropic yield (afterCotecchia & Chandler, 1997).

6

4

2

0

-2

-40 2 4 6 8

p'/p'

e = 0.6

q/p

' e

e

Final intrinsicstate boundarysurface

e=

1.60e

=1.70

e=

1.80

Outer stateboundarysurface

B

Post-yield contoursof void ratio

Figure 11.10. Behaviour of Bothkennar clay normalized with respect to p�e (from Smith et al., 1992).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 156

curve of the material. As indicated by Leroueil & Vaughan (1990), destructuration can be obtained bycompression outside the limit state curve, by shearing or by swelling (Figure 11.11).

11.3.1 Destructuration by compressionDestructuration by compression has been shown for soft clays by Leroueil et al. (1979) and Tavenas &Leroueil (1985). The effect of removing microstructure was examined by unloading test specimens afteranisotropic consolidation to axial stresses in excess of the preconsolidation pressure and testing them inanisotropic compression under constant K � �r / �a ratio or in undrained conditions. The behaviour wasthen compared with results obtained from the intact microstructured soil tested after consolidation tosimilar overconsolidation ratios. Figure 11.12 shows typical undrained triaxial test results. The destruc-tured clay is less stiff and reaches a smaller peak strength at a larger failure strain. The resulting failureenvelopes for the destructured clays were thus lower than those for the microstructured clays. As previ-ously indicated, similar results have also been shown by Amorosi & Rampello (1998) (Figure 11.4).

157

(σ ' + σ' )/2

Swelling yield

φ' line

φ' line

Shearing yield

Compre

ssio

nyi

eld

Shearing yield

a r

(σ'

-σ'

)/2

ra

Figure 11.11. Schematic figure showing different possible zones of yielding (from Leroueil & Vaughan, 1990).

Figure 11.12. Stress-strain relationships from undrained triaxial tests on intact and destructured clays (afterTavenas & Leroueil, 1985).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 157

Destructuration by compression can also be observed in weak mudrocks (Banks et al., 1975;Leroueil & Vaughan, 1990) and in clay shales (Picarelli, 1991b). Figure 11.13 illustrates well the phe-nomenon of progressive destructuration. The Laviano clay shale was cyclically loaded and unloaded inan oedometer, with maximum stress increasing at each cycle. It can be seen that the swelling indexCs � �e/�log � increases with applied stress and accumulated strain to approach the Cs value of theremoulded and reconstituted soil.

It is worth noting that destructuration by compression can also be reached by drying the soil, asshown in the heavily overconsolidated Todi clay (Calabresi & Scarpelli, 1985; Rampello, 1989, 1991).Figure 11.14a shows the strength envelopes obtained for the intact soil and for the soil first air dried andthen left to swell before shearing. The strength envelope obtained after drying is well below the strengthenvelope of the intact soil, indicating some destructuration. To take into account the influence of voidratio on strength envelopes, Rampello (1989) normalized them with respect to p�e. The normalizedstrength envelope of the undisturbed soil is then only slightly above that of the soil after drying-wetting,indicating that interparticle bonding is existent but not strong in Todi clay.

11.3.2 Destructuration by shearingDestructuration may also occur by shearing. This can be illustrated by triaxial tests performed by Hightet al. (1997) on Crag sand, a very dense, often shelly, marine sand. These authors performed undrainedtriaxial compression and extension tests with unload-reload loops at increasing levels of strain. As shown

158

Figure 11.13. Results of oedometer tests on Laviano clay shale (from Picarelli, 1991b).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 158

in Figure 11.15 for compression tests, as the previous maximum axial strain increases, the secant mod-ulus decreases, demonstrating destructuring.

In general, it is relatively difficult to quantify the effect of destructuration by shearing because of the possible formation of shear discontinuities in triaxial testing, and with possible orientation of parti-cles (see Section 8). One way of inducing homogeneous shear strains in soils is by sampling them withtubes driven or jacked into the ground. During sampling, the soil is subject to a compression-extension-compression cycle of loading (Baligh, 1985). These aspects are described by Hight & Leroueil (2002*);however, a few examples can be considered to illustrate the effects of destructuration by shearing.Figure 11.16a compares CIU test results performed on samples of clay taken by 5 cm diameter tube andhand-cut block. The effect of destructuration is similar to that observed in Figure 11.12 where the soilwas destructured by compression. Figure 11.16b compares oedometer compression curves obtained onsamples taken by blocks and with an 85 mm sampler. Destructuration increases the recompression index

159

UndisturbedAfter free swellAfter dryingReconstituted

1.0

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.0

p' (MPa)

q(M

Pa)

UndisturbedAfter free swellAfter dryingReconstituted o.c.Reconstituted n.c.

1.2

0.8

0.6

0.4

0.2

00 0.2 0.4 0.6 0.8 1.2

q/p e

1.0

1.0

p'/p'e

(a) (b)

'Figure 11.14. Failure envelopes for undisturbed, free swollen and dried-wetted samples of Todi clay: (a) in an usualstress diagram; (b) normalized with respect to the equivalent pressure at failure, p�e (from Rampello, 1989, 1991).

Figure 11.15. Progressive destructuring in rotary cored samples of Crag sand by unload-reload compression cycles(from Hight et al., 1997).

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Cs, decreases the compression index Cc and decreases the preconsolidation pressure, which is less welldefined. Figure 11.16c, which compares limit state curves obtained on high quality and low qualitysamples, demonstrates that the entire limit state curve shrinks due to destructuration induced by sam-pling and shearing.

Loss of microstructure may also result from the shearing accompanying deep compaction of sands(Mitchell & Solymar, 1984).

11.3.3 Destructuration by swellingIn some materials in which the inter-particle bonds are not strong enough to resist the swelling forcesgenerated when the effective stresses are reduced, there can be destructuration by swelling as indicatedin Figure 11.14 for Todi clay. Figure 11.14a shows the strength envelopes obtained on the undisturbedmaterial and after free-swelling, i.e. after 12% expansion. It can be seen that the strength envelope ofthe undisturbed soil is much higher than that of the same material destructured by swelling. As fordestructuration by drying, when the same results are normalized with respect to p�e (Figure 11.14b), thestrength envelope of the undisturbed soil remains above the other envelopes but the difference, reflectingthe effect of microstructure only, is smaller. As shown by Leroueil & Vaughan (1990) and Hight et al.(2002a*), London clay is also prone to destructuration by swelling.

Also indicated on Figure 11.7 is the fact that Go can decrease at small effective stresses in compari-son with the Go– � relationship of the intact soil, if the material is subjected to destructuration byswelling. Some destructuring can also be observed in Figure 11.6 for the cemented sand unloaded andthen reloaded.

160

Figure 11.16. The effects of shear from sampling disturbance on strength, on the abruptness of yield and on yieldstress: (a) undrained triaxial tests (from Lefebvre, 1970); (b) oedometer tests (from Holtz et al., 1986); (c) limit state(after La Rochelle et al., 1981).

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11.3.4 Destructuration by weatheringWeathering may also contribute to destructuration of geomaterials. It is generally associated with cli-matic phenomena (wetting-drying cycles, freeze-thaw cycles) or chemical action. It may produce adegradation of the bonds between particles or aggregates and thus a destructuration of the materialresulting in a reduction of strength, as indicated in Table 11.1 from Taylor & Cripps (1987). Nishi et al.(1989) examined the variation of the small strain shear modulus, Go, of three different soft rocks pro-gressively weathered by freeze-thaw cycles, drying-wetting cycles or natural outdoor weathering.Figure 11.17 shows typical results. It can be seen that Go decreases as the weathering process prog-resses, in a manner very similar to that shown in Figure 11.7.

11.3.5 Destructuration by fatigueYoshinaka & Osada (1995) examined four soft sedimentary rocks under static and cyclic loading. Themonotonic tests were performed in undrained conditions at an axial rate of 0.01 mm/min. In the fatigue(or cyclic) tests the axial load was repeatedly applied at a frequency of 0.2 Hz until the specimenreached failure. The normalized maximum deviatoric stress, defined as the ratio of the maximum devi-atoric stress applied in cyclic tests to the deviatoric stress at failure in the monotonic tests, varied from80% to 120%. Figure 11.18 shows the normalized maximum deviatoric stress applied in the fatigue testas a function of the number of cycles necessary to reach failure for the Yokohama siltstone. It can beseen that the number of cycles to failure increases as the applied stress level decreases and that failureat the first loading cycle is reached at a normalized maximum deviatoric of about 1.2. This latter ratiois explained by the fact that strain rates in cyclic loading tests are much larger than those in monotonictests. It thus appears that the strength under cyclic loading is influenced by strain rate, as evidencedunder the first loading cycle, and by fatigue during the subsequent loading cycles. Also, on the basis ofmeasured pore pressures, Yoshinaka & Osada (1995, 1998) concluded that the reduction in strengthresults from a lowering of the strength envelope with the accumulation of cycles and time.

161

Table 11.1. Strength characteristics of selected fresh and weathered mudrocks andoverconsolidated clays (after Taylor & Cripps, 1987).

Fresh Weathered

�� (°) c� (kPa) � (°) c� (kPa)

London clay 20–29 31–252 17–23 1–18Bearpaw shale 25–30 10–152 20–28 0–41Lower Oxford clay 23–40 10–216 21–28 0–20Upper Lias clay 25 27 18–25 1–17Keuper Marl �40 �30 25–42 2–80Coal Measures Mudrock 46 131 26–39 0–25

1.0

0.5

0.10.1 0.5 1.0 5.0 10

Effective confining pressure σ ' (kgf/cm )2c

Non-weathered

Weathered

G(b

yu

ltra

son

icp

uls

ete

st)

(x1

0kg

f/cm

)4

o2

Pro

gre

sso

fw

ea

the

ring

Tuff

Figure 11.17. Variation of Go with the confining stress at different degrees of weathering (after Nishi et al., 1989).

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Lefebvre & Leboeuf (1987) performed undrained triaxial tests on eastern Canada clays under monot-onic conditions at different strain rates and under cyclic conditions (Figure 11.19) and came to conclu-sions similar to those of Yoshinaka & Osada (1995): (a) cyclic behaviour is influenced by strain rate; (b)due to fatigue, there is a lowering of the strength envelope of the overconsolidated material as the num-ber of cycles increases. This latter aspect was confirmed by Santos et al. (1997), on a residual soil fromRio de Janeiro.

As indicated by several authors, the destructuration process is probably associated with accumulationof plastic strains (Gens & Nova, 1993; Kavvadas & Amarosi, 1998). This means that the behaviour of structured soils should probably be examined in terms of strains rather than in terms of stresses.Unfortunately, the strains involved in the pre-yield range are small, difficult to measure, and with an ori-gin that is generally not clearly defined.

11.3.6 Conclusion on destructurationFrom previous paragraphs, it appears that the effects of destructuration are similar whatever the modeof destructuration: compression, shearing, swelling, weathering or fatigue. They can be summarized as

162

s = 0.5 MPas = 2.0 MPa3

3

1 10 10 100

0.5

1.0

1.5

2 3

Number of cycles to failure (N)

No

rma

lize

dm

axi

mu

md

evi

ato

rst

ress

Yokohama siltstone

Figure 11.18. Number of cycles to failure as a function of the maximum deviator stress applied in undrained cyclictraixial tests (from Yoshinada & Osada, 1995).

Extrapolated strengthfor the first cycle

Rate effect

1 2 310 10 10Number of cycles

0.2

0.4

0.6

0.8

1.0

(σ−

σ)

2σ

'a

rm

ax

vc

0.35

0.46

Cyclic strengthMonotonic strength (0.5%/h)Extrapoled from monotonic test at0.5%/h for a strain rate equivalentto a frequency of 0.1 Hz

Figure 11.19. Cyclic strength of normally consolidated B6 clay (modified after Lefebvre & Leboeuf, 1987).

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follows: (a) decrease in stiffness of the soil inside the limit state curve; (b) decrease in peak shearstrength and in preconsolidation pressure, as well as a shrinkage of the entire limit state curve; and (c) decrease in compression index.

11.4 Structuration and restructuration with time (aging)

There is evidence that, in some materials, microstructure may develop or may recover, at least to someextent, with time.

Leonards and Altschaeffl (1964), Perret (1995) and Leroueil et al. (1996) showed that, in youngreconstituted clays, the development of preconsolidation pressure during secondary consolidation canbe much higher than that related to the decrease in void ratio. For reconstituted Jonquière clay consoli-dated to 10 kPa (Figure 11.20a) and left under this applied stress, the preconsolidation pressure wasequal to 18.5 kPa after 120 days whereas it should have been 11.5 kPa on the basis of the change in voidratio only. It is worth noting that Burland (1990) reported void ratio-effective overburden stress rela-tionships for normally consolidated deposits (sedimentation compression line) at void ratios higher thanthe compression line obtained in the laboratory on reconstituted clay (see Section 18.3 and Figure 18.2),indicating development of microstructure during the formation process of sedimentary clays.

Schmertmann (1991) presented data of a 20 cm diameter rigid plate loaded over a dry clean and uni-form sand. Figure 11.21 shows the plate settlement versus plate load. It can be seen that after 844 minof rest under an approximate constant load, the sand had acquired a resistance to further loading well inexcess of what could be expected on the basis of the settlement (or change in void ratio) observed dur-ing that time. Other evidence of strength increases with time have been reported by Schmertmann(1991), and by Mitchell & Solymar (1984) who observed sand restructuration with time after destruc-turation by deep compaction.

Due to their viscous behaviour, soils have a tendency to rearrange their particles and generallydecrease their void ratio under constant effective stress. They may also be subjected to structuring phe-nomena that have a tendency to fix particles into position. The rate of strain determines which of thetwo opposing mechanisms would dominate. At relatively high strain rates the former mechanism willdominate as described in Section 9. At relatively low strain rates, structuring effects may become impor-tant and modify the strain rate effects. For example, on Batiscan clay, Leroueil et al. (1985a) observedcompression curves from CRS oedometer tests that were well organized, according to the stress-strain-strain rate model (Equations 9.1 to 9.3), for strain rates larger than 10�7s�1 (Figure 11.22). However, as

163

2.6

2.4

2.2

2.0

1.8

Vo

idra

tioe

4 5 6 7 8 9 2 310

4 5 6 7 8 9 3 2 3 4 510

σ ' (kPa) G (kPa)v o

α(b)

α

120 d

(a)

Figure 11.20. Compression curve (a) and variation of the maximum shear modulus Go with void ratio (b) for artificially sedimented Jonquière clay (from Perret, 1995 and Leroueil et al., 1996).

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can be seen on the figure, the test performed at the strain rate of 1.69 � 10�8s�1does not fit with theother results, the compression curve crossing the compression curves obtained at larger strain rates.This is attributed to structuring phenomena that become dominant at this latter strain rate. Marques et al. (2002) observed similar phenomena on another eastern Canada clay; they also observed that struc-turing effects become more important as temperature increases from 10° to 50°C.

164

0.3

0.4

0.5

0.6250 300 350 400

9

98

97

8

UF K-boxK = active

Load on 8" diameter plate (lbs)

Pla

tese

ttle

me

nt

(in)

12 min

844 minaging

Dry medium sand

Figure 11.21. Aging-preconsolidation effect demonstrated by plate load test on sand (from Schmertmann, 1991).

Effective stress σ' (kPa)v

ε v1

εv2

εv3

εv4

ε v5

ε v6

= 1.43 x 10

= 5.30 x 10

= 2.13 x 10

= 5.33 x 10= 1.07 x 10= 1.69 x 10

-5

-6

-6

-7

-7

-8

-1CRS tests (s )

Batiscan

0 50 100 150 200 2500

5

10

15

20

25

100

50

0

Str

ain

ε v (

%)

Exc

ess

pore

pres

sure

² ∆u b

(kP

a)

εv4 ε v5 ε v6, and

Figure 11.22. Typical CRS oedometer tests on Batiscan clay (from Leroueil et al., 1985a).

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12 INFLUENCE OF DISCONTINUITIES

This Section is mostly concerned with clayey soils and soft rocks in which the initially intact materialhas been sheared into elements due to different processes: unloading, tectonic activity, ice-thrusting,etc. These processes result in fissured stiff clays, stiff jointed clays, clay shales, tectonized clay shales,etc. These materials are then made up of intact elements, often microstrutured, separated by disconti-nuities such as fissures, joints or faults. These discontinuities influence their mechanical behaviour, notso much in terms of compressibility but mostly in terms of stiffness and strength. The main influencingfactors are: mechanical properties of the intact material; density of discontinuities compare to the scaleof the considered problem (size of specimens in the laboratory or of a natural slope in situ); previousrelative displacement between adjacent elements; and direction of shearing compared to the orientationof discontinuities. The general behaviour of these geomaterials has, in particular, been described bySkempton & Petley (1967), Skempton (1977), D’Elia et al. (1998), and Picarelli et al. (2000, 2002*).

12.1 Stiff fissured and jointed clays

The study made by Calabresi & Manfredini (1973) on the stiff jointed Santa Barbara clay illustrateswell the influence of discontinuities on the strength of clays. In addition to well defined bedding planes,the Santa Barbara stiff clay deposit is subdivided by several families of joints and by a series of faultswith a shear zone up to 20–30 cm thick.

Direct shear tests were performed with shear planes in the intact soil and along joints. Figure 12.1shows shear stress-displacement curves for an effective normal stress of 100 kPa. The intact clay showsvery brittle behaviour. On the other hand, shearing along a joint shows almost no peak. It is worth not-ing, however, that in this latter case, failure is reached at a very small displacement. Figure 12.2a showsthe corresponding strength envelopes. The peak strength envelope along joints has the same frictionangle as the intact clay, but a much smaller cohesion. Calabresi & Manfredini (1973) suggested that thiscohesion could be due to the roughness of the joint surface. No difference appears in the residual val-ues of strength, but the displacement required to reach residual conditions is smaller along a joint thanfor the intact material.

Along a fault, stress-displacement curves presented very small peak or no peak at all with a strengthvery close to the residual strength of the clay (Figure 12.2b).

The behaviour of the Santa Barbara clay was related to the surface characteristics of the discontinu-ities: large displacements occurred along faults, the surface appeared polished and slickensided, and theresidual strength had already been reached; along joints (and bedding planes), no displacement had

165

Figure 12.1. Stress displacement curves obtained in multiple stage direct shear tests on Santa Barbara clay, �n � 100 kPa (after Calabresi & Manfredini, 1973).

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occurred previously and the surface appeared rough. As a consequence, the strength was characterizedby the same friction angle as the intact soil but almost no cohesion. It corresponds to the “post-rupture”strength indicated in Section 8.3. Backanalyses of instabilities confirmed the strength parameters foundin the laboratory study.

These views were reinforced by studies made on London clay. Skempton (1977) found that thestrength of joints and fissures in London clay (c� � 0; �� � 20°) is smaller than the peak strength of the intact material (c� � 7 kPa; �� � 20°), larger than the residual strength (c� � 0; ��r � 13°) and verysimilar to the “fully softened” or “critical state” strength obtained on remoulded, normally consolidatedclay (Figure 12.3).

12.2 Tectonized clay shales

Italian researchers, in particular Picarelli et al. (2000, 2002*), described the behaviour of so-called tec-tonized clay shales, clay shales that have been subjected to large strains and displacements during majortectonic events. The fissures materials are characterized by fine grained indurated soils crossed by anetwork of small polished or slickensided fissures. These are spaced from a few millimetres to a fewcentimetres and subdivide the material into very small fragments. The water content is lower than theplastic limit, with a liquidity index typically between �0.2 and �0.4. When softened due to unloadingand swelling, or weathering, these fissured clay shales become a new material made up of small litho-relicts of the original formation mixed with a softer clay matrix. In such materials, the overall water con-tent results from that of the lithorelicts with negative liquidity index and that of the matrix with aliquidity index which can be close to 1.0. The mechanical behaviour of these latter softened clay shales

166

Figure 12.2. Strength envelopes for the Santa Barbara clay: (a) in the intact material and along a joint at elevation110 m; (b) along a fault at elevation 125 m (from Calabresi & Manfredini, 1973).

Figure 12.3. Strength of fissures and joints compared with critical state strength for London clay (after Skempton,1977).

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is also summarized by Picarelli et al. (2000, 2002*) and is not described here. Only aspects of thebehaviour of “intact” tectonized clay shales will be presented hereunder.

Olivares et al. (1993) presented results of drained and undrained triaxial tests performed on theBisaccia tectonized clay shale in both undisturbed and reconstituted conditions. For the reconstitutedclay, the soil was isotropically consolidated under a maximum effective stress of 500 kPa and thenunloaded before shearing. Figure 12.4 shows the strength envelopes, normalized with respect to theequivalent consolidation pressure p�e determined on the isotropic consolidation line of the reconstitutedclay. The strength envelope of the undisturbed material is much lower than the strength envelope of thesame soil reconstituted, which has been attributed to the fissures along which the shear strength is closeto residual.

Picarelli et al. (2000) indicate two main stages during shearing of tectonized clay shales: a first stage,in which the local strength (i.e. the shear strength along fissures) is not completely mobilized. The spec-imen then remains uniform and the stress path followed in an undrained test during this stage is essen-tially at p� � cst; a second stage, in which the local shear strength is attained and the mechanism of soildeformation is characterized by important plastic “strains” due to slipping along fissures. D’Onofrio et al. (1998) examined the behaviour of Bisaccia clay, both undisturbed and reconstituted, in the pre-failure range. Performing cyclic and resonant column torsional shear tests at increasing strain levels,they found the elastic threshold strain to be typically 3 times larger for the reconstituted material thanfor the undisturbed one. In fact, in comparison with other experimental data (Figure 4.16), the reconsti-tuted results are similar whereas the tectonized clay shales give smaller threshold strain values.

The presence of hard fragments in tectonized clay shales also influences the measured residualstrength. As shown in Figure 12.5, under a given normal effective stress, the undisturbed soil has alarger residual friction angle than the reconstituted material, especially under small normal effectivestresses.

12.3 Fissured and jointed soft rocks

The stiffness of rock masses is strongly influenced by discontinuities (Cook, 1992). This was demon-strated for chalk by Clayton et al. (1994, 2002*). These authors compared the mass and intact stiffnessof chalk from three sites with different porosities. The rock at each site was highly fractured with typi-cal sub-horizontal discontinuity spacing of between 10 and 160 mm. The mass stiffness was determinedusing surface wave geophysics; the intact stiffness was measured in the laboratory, in triaxial tests, onlocally instrumented specimens.

Figure 12.6 shows intact and mass stiffness for the three sites. The intact stiffness is strongly depen-dent on porosity and density. The mass stiffnesses are much lower, but similar for the three sites, whichwas explained as follows by Matthews et al. (2000): The high-density chalk displayed a loose fracture

167

Figure 12.4. Bisaccia clay: strength obtained by drained and undrained triaxial tests (from Olivares et al., 1993;Picarelli et al., 2002*).

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block system resulting in the rock mass stiffness being less than 20% of the intact rock stiffness. Thelow-density chalk had a tight fracture block system resulting in a much smaller difference betweenintact and mass stiffness.

13 INFLUENCE OF SOIL WATER CHEMISTRY

It is known that pore fluid composition influences the behaviour of active clays (Mesri & Olson, 1970,1971; Sridharan, 1991, 2001; Mitchell et al., 1973; Barbour & Fredlund, 1989; Di Maio, 1996a, b; Di Maio & Onorati, 2000). The following sections will briefly illustrate possible effects.

168

Figure 12.5. Residual shear strength of Laviano tectonized clay shale (from Picarelli, 1991; Picarelli, 2000).

Figure 12.6. Mass and intact stiffness for chalks with different intact dry density (after Clayton et al., 1994;Clayton et al., 2002*).

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13.1 Isotropic and one-dimensional compression

Figure 13.1a shows isotropic and one-dimensional compression curves for the Bisaccia clay (a smectitesoil) reconstituted at a water content slightly larger than the liquid limit with distilled water and with a1 M NaCl solution (Di Maio, 2002). The influence of water composition is important, the compressioncurves with distilled water being at much higher void ratio. It is worth noting that the opposite holds forsome clays such as Norwegian and eastern Canada clays: for these materials, the sedimentation curveobtained in fresh water is at void ratios that are smaller than those of the sedimentation curve obtainedin a marine environment (Bjerrum, 1967; Locat, 1982); (see Figures 11.2b and 18.8).

Picarelli et al. (2000, 2002*) illustrated the consequence of an active marine clay being exposed tofresh water for the Bisaccia clay. The clay, reconstituted with 1 M NaCl was first one-dimensionallycompressed to high stresses and then unloaded to a vertical stress of 10 kPa. Finally, when exposed todistilled water under the same vertical stress, the soil specimen swelled by 34% of its volume.

13.2 Peak and critical state strength envelopes

Di Maio & Onorati (2000) and Di Maio (2002) performed undrained and drained triaxial compressiontests as well as direct shear tests on the same Bisaccia clay reconstituted and immersed with distilledwater or with 1 M NaCl solution. Figure 13.1b compares the peak strength envelopes deduced fromundrained compression tests on normally consolidated specimens. It can be seen that the influence ofpore fluid composition is very important, the friction angle dropping from about 18° with NaCl solu-tion to about 10° with distilled water.

13.3 Residual strength

Di Maio (1996a, b) showed that the effect of pore water chemistry can also be important on residualfriction angle. Figure 13.2 shows results obtained on three southern Italy clays and on Ponza bentonite

169

Figure 13.1. Influence of pore water chemistry on the behaviour of reconstituted Bisaccia clay: (a) Intrinsic compression lines; (b) CIU triaxial test results on the normally consolidated material (after Di Maio & Onorati,2000; Di Maio, 2002).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 169

prepared either with saturated NaCl solution or with distilled water. It can be seen that, with the excep-tion of Marino clay, ��r changes from about 14° with the saturated NaCl solution to 6° with distilledwater, which may be extremely important from a practical point of view. Figure 13.3 shows the varia-tion of ��r with concentration of NaCl solution for Ponza Bentonite; most of the change in ��r occurs forconcentrations between 0 and 35 g/l NaCl, which should be an encouragement for evaluating residualfriction angle in laboratory with pore water having a chemistry close to that existing in situ.

14 INFLUENCE OF PARTIAL SATURATION

14.1 Soil-water characteristic curve

The soil-water characteristic curve (SWCC), also called the water retention curve, is the relationshipbetween the amount of water in the soil (gravimetric or volumetric water content or degree of satura-tion) and matric suction (ua � uw) in the soil, called herein suction. It gives an idea of the pore size distribution of the considered soil and important information concerning the hydraulic as well as the

170

Figure 13.2. Residual shear strength against normal stress of specimens reconstituted with water and of specimensreconstituted with saturated NaCl solution (from Di Maio, 1996a).

Figure 13.3. Residual friction angle of Ponza bentonite under normal stresses of 250–400 kPa for various concentrations of NaCl solution (after Di Maio, 1996b).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 170

mechanical characteristics and behaviour of soils when unsaturated. Vanapalli (1994) and Sillers et al.(2001) define three stages during the process of desaturation (increasing suction; see Figure 14.1):

– Capillary saturation zone: The pore water is in tension in this zone, however, the soil remains essen-tially saturated due to capillary forces. This zone ends at the air entry value or bubbling pressure(ua � uw)b where the air starts to enter in the largest pores (Figure 14.1).

– Desaturation or funicular zone: water within the pores is increasingly displaced by air in the desatu-ration zone. The desaturation zone ends at the residual water content where the liquid phase becomesdiscontinuous.

– Zone of residual saturation: the water is tightly held to the soil. Increasing soil suction does not resultin significant changes in water content and moisture movement occurs mainly as vapor flow. The suc-tion at which all soils approach zero water content would be approximately 1 000 000 kPa (Croney &Coleman, 1961; Fredlund & Xing, 1994).

Figure 14.2 shows SWCC for different soils. The SWCCs illustrate the influence of the pore size dis-tribution of the soil, the air entry value being about three orders of magnitude larger for the Regina claythan for the sand. From capillary theory, the radius of curvature of a water surface is related to thematric suction as follows:

r � C/(ua � uw) (14.1)

where C is a constant equal to 2Tscos �/�w � 150 � 10�3N/m where Ts is the surface tension of water, � is the angle of contact between soil particles and air-water interface and �w is the unit weightof water. The air-entry pressure values that can be deduced from Figure 14.2 would thus correspond tothe largest entrance pore radius in the order of 0.035 mm for the sand and 0.05 �m for Regina clay.

The SWCC is defined by progressive drying of an initially saturated soil (ABC in Figure 14.3). Uponwetting, there is a hydraulic hysteresis (CDA in Figure 14.3). If wetting is relatively rapid, some air(residual air content) is trapped in the soil and the wetting curve goes towards a degree of saturation,called submergence degree of saturation, Srf, that is smaller than 1.0 (see dashed line in Figure 14.3;also see Rojas, 2002). It is only with time, by diffusion of air, that the degree of saturation will pro-gressively increase from Srf to 1.0.

171

Figure 14.1. Soil-water characteristic curve illustrating the regions of desaturation for a soil (from Sillers et al.,2001).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 171

The two curves ABC and CDA are limits of an accessible domain for any cycle of drying and wetting (e.g., 12341 in Figure 14.3). Buisson and Wheeler (2000) explained hysteresis as follows(Figure 14.4): On wetting, the radius of the meniscus separating air and water increases as suctiondecreases (see Equation 14.1) and the meniscus migrates inwards from (1) to (2) in Figure 14.4a. Assuction decreases, further and the radius of the meniscus continues to increase, a limiting condition isreached (3) when the radius is equal to that of the largest sphere that can fit within the void. Any furtherreduction of suction would result in the void flooding with water. The value of suction at which this voidfloods with water during a wetting path is therefore governed by r3. On drying, the radius of the menis-cus separating air and water decreases and the meniscus migrates inwards from an initial position (4)towards (5) in Figure 14.4b. A critical point is reached with the radius r5 for which any further increasein suction would result in the void filling with air. Hydraulic hysteresis thus results from the fact thatthe critical radius for emptying of the void, r5, is smaller than the critical radius for flooding the void, r3.

In situ, soils may be subjected to drying-wetting cycles such as 1–2–3–4–1 in Figure 14.3 and could beat point E at a given time. Consequently, suction in soil cannot be defined on the basis of water contentor degree of saturation only.

Numerous equations have been proposed in the literature to represent SWCC. Reviews are providedby Fredlund & Xing (1994), Leong & Rahardjo (1997) and Sillers et al. (2001).

172

Figure 14.2. Typical soil-water characteristic curve for four Canadian soils (from Vanapalli et al., 1999).

Figure 14.3. Elasto-plastic representation of hydraulic hysteresis in a rigid soil.

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The SWCC of a soil reflects its pore size distribution. It thus varies with the void ratio and the fabricof the soil. This can be illustrated by the following examples:

– Kawai et al. (2000) determined SWCC for clay specimens statically compacted at different voidratios. As shown in Figure 14.5, the air entry value increases from 5 kPa to 83 kPa when the void ratiodecreases from 2.1 to 0.9.

– Even at about the same void ratio, a given soil may have different fabrics and thus different SWCCs.This is evident in Figure 14.6 showing very different SWCCs for a non-plastic glacial till compactedat degrees of saturation smaller and larger than the corresponding value at the optimum condition. Itcan be seen that the air entry value is about one order of magnitude larger when the soil is compactedwet of optimum. Similar results were obtained by Vanapalli et al. (1999). Therefore, except for coarseand clean materials, SWCC should not be seen as an intrinsic characteristic of a soil.

173

Figure 14.4. Mechanism for flooding and emptying of a void (after Buisson & Wheeler, 2000).

Figure 14.5. Relation between air entry value and void ratio for a clayey soil (after Kawai et al., 2000).

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14.2 Strength of unsaturated soils

A number of researchers have attempted to extend the principle of effective stress to unsaturated soils,with the objective to describe soil behaviour as a function of a unique stress variable. Several expres-sions have been suggested for a generalized effective stress, the best known being that proposed byBishop (1959):

� � ( � ua) � �(ua � uw) (14.2)

where � total stress, uw � pore-water pressure, ua � pore-air pressure, ( � ua) � net total stress,(ua � uw) � s � matric suction, and � � effective stress parameter which has a value of 1 for saturatedsoils and a value of 0 for dry soils.

When Equation 14.2 is applied to the strength of unsaturated soils, the Mohr-Coulomb criterionbecomes:

�f � c� � {( n � u a) � �(ua � u w)}tan �� (14.3)

where c� � effective cohesion (same as for saturated soil) and �� � effective friction angle (same as forsaturated soil).

In general, unsaturated soils have a shear strength that is larger than that of the same soil saturated.Fredlund et al. (1978) proposed that it could be expressed as a linear combination of the net normalstress ( n � ua) and of the matric suction (ua � uw).

�f � c� � ( n � ua)tan �� � (ua � uw)tan �b (14.4)

where tan �b � strength parameter associated with matric suction.Equation 14.4 is schematically represented in Figure 14.7. The term (ua � uw)tan �b appears as a

cohesion due to suction.Experimental evidence shows that Equation 14.4 is valid for a variety of unsaturated soils, but, most

of the time, within limited ranges of matric suction and net normal stress. Also, limitations on the useof Equation 14.4 have been found:

a) Escario & Saez (1986), (Figure 14.8), Gan & Fredlund (1988), De Campos & Carillo (1995), andOloo & Fredlund (1996), have shown that at suctions smaller than the air entry value, whereas thesoil remains saturated, �b is equal to ��. In these conditions, a change in suction is equivalent to achange in effective stress.

b) Delage et al. (1987) and Maâtouk et al. (1995) observed strength envelopes for silts obtained at dif-ferent suctions which were converging towards a unique point, indicating that there is no effect ofsuction for net normal stress in excess of a given value.

174

Figure 14.6. Soil-water characteristic curves of LG2 till compacted at different conditions (after Watabe et al.,2000).

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c) It is known that in some cohesionless soils shear strength first increases and then decreases as suc-tion is progressively increased. Examples of such behaviour are reported by Donald (1956), Escario& Juca (1989) and Fredlund et al. (1996). An example is shown in Figure 14.9.

Equation 14.4 and its previously mentioned limitations result from the observation of soil whensheared at different net normal stresses and suctions. An apparently more rational approach has beenproposed by Fredlund et al. (1996) and Vanapalli et al. (1996). The main features are presented in thefollowing paragraphs.

As indicated by Fredlund et al. (1995), the contribution of matric suction to shear strength of anunsaturated soil can be assumed to be proportional to the product of matric suction, (ua � uw), and thenormalized area of water, aw:

� � aw (ua � uw)tan �� (14.5)

where aw is the ratio of the area of water corresponding to any degree of saturation to the total area ofwater at saturation.

175

Figure 14.7. Shear strength of unsaturated soils (after Fredlund et al., 1978).

Figure 14.8. Shear strength as a function of suction for different net normal stresses. Direct shear tests on Madridclayey sand (from Escario & Saez, 1986).

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The normalized area of water, aw, presents similarities with the normalized volumetric water content�, and Fredlund and co-workers proposed the following relationship:

aw � � (14.6)

where

(14.7)

in which �, �s and �r are the volumetric water content respectively at the considered suction, at satura-tion and at the residual condition. When �r is equal to zero, � is also the degree of saturation.

is a soil parameter dependent upon soil type. According to Fredlund et al. (1996), is close to 1 insandy soils, but can be greater than 1 when plasticity increases.

The strength of unsaturated soil can then be written as:

�f � c� � ( n � ua)tan �� � (ua � uw)� tan �� (14.8)

Fredlund et al. (1996) consider that � is a function of matric suction and can be deduced from theSWCC. As a result, the required parameters to determine the strength of an unsaturated soil would bethe strength parameters c�and �� of the saturated soil and the soil-water characteristic curve. It shouldbe emphasized that Equation 14.8 explains that �b � �� at suctions smaller than the air entry value anda possible decrease in shear strength when suction increases in excess of a given value, as shown inFigure 14.9. An application of Equation 14.8 to a decomposed granite from Hong Kong is shown inFigure 14.10. Figure 14.10a shows the soil-water characteristic curve for this soil and the calculatedvariation of shear strength with suction is shown in Figure 14.10b, assuming a value of 1.0.

When � 1 and �r � 0, � becomes equal to Sr and Equation 14.8 becomes equivalent to Equation14.3 with � � Sr. Fleureau et al. (1995) and Öberg & Sällfors (1997) showed that Equation 14.3 givesgood agreement with the results for sands.

On the basis of shear strength data obtained on 14 different soils, Khalili and Khabbaz (1998)observed that � in Equation 14.3 can be empirically related to the suction ratio {(ua � uw)/(ua � uw)b}where (ua � uw)b is the air entry value (see Figure 14.11). Khalili & Khabbaz (1998) specified that thelower bound values in Figure 14.11 correspond to coarse-grained materials whereas the upper boundvalues correspond to fine-grained materials.

It is worth noting that the shear strength of saturated soils is characterized by three different levels:the peak strength envelope, upper part of the limit state curve; the strength envelope of the normallyconsolidated soil, or critical state line; and the residual strength envelope (see Figure 8.2). At the presentstate of understanding of unsaturated soil mechanics, these differences do not clearly appear althoughthey certainly should.

� �� � �

� � �r

s r

176

Figure 14.9. Results of direct shear tests on fine Frankston sand (data from Donald, 1956; from Fredlund et al.,1996).

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177

Figure 14.10. Behaviour of a completely decomposed tuff from Hong Kong: (a) soil-water characteristic curve; (b) shear strength versus suction (from Fredlund et al., 1996).

Figure 14.11. Relationships between the effective stress parameter � and suction ratio (ua � uw) /(ua � uw)b (fromKhalili & Khabbaz, 1998).

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14.3 Compression curves of unsaturated soils

Bishop (1959) proposed the use of the generalized effective stress given by Equation 14.2 to describethe behaviour of unsaturated soils in both compression and shear. The strength is then defined byEquation 14.3. The possible use of this generalized effective stress for characterizing the compressibil-ity of soils in general appears more difficult. Jennings & Burland (1962) illustrated this fact. Theseauthors tested a number of samples of silt that were air dried from slurry. The hatched zone in Figure 14.12shows the domain occupied by the compression curves of the air-dried soil. The full line on the samefigure is for a sample soaked under zero stress and then compressed. The curve of the air-dried soil liesabove that of the soaked sample. Some of the air-dried samples were loaded to applied pressures of 200,

178

Figure 14.12. Oedometer compression curves on air dry and saturated silt showing the effects of soaking (afterJennings & Burland, 1962, and Burland, 1965).

Figure 14.13. Isotropic compression curves at essentially constant suction performed on speswhite kaolin (afterWheeler & Sivakumar, 1995).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 178

400, 800 and 1600 kPa and then soaked. As seen in Figure 14.12, they collapsed and came to equilib-rium (full triangles) at a void ratio very close to the compression curve of the initially soaked soil. Someother air-dried samples were loaded to the same applied pressure values. They were then soaked withthe applied stress being reduced in an effort to maintain a constant void ratio. It can be seen that thesamples come to equilibrium (full circles) on the soaked line. Soaking corresponds to a decrease in suc-tion and then, according to Equation 14.2, to a decrease in effective stress. On the basis of classical soilmechanics, a decrease in effective stress should be associated with an increase in void ratio of the soil.The reverse is observed, indicating that the behaviour of partly saturated soils cannot in general bedescribed by effective stress. From such results, it has been stated that the stress state of unsaturatedsoils has to be described by two stress state variables, usually ( � ua) and (ua � uw).

The effect of matric suction on the compression of unsaturated soils has been studied by manyresearchers (Matyas & Radhakrishna, 1968; Wheeler & Sivakumar, 1995; Leroueil & Barbosa, 2000;and many others). Figure 14.13 shows isotropic compression curves obtained in constant suction testsperformed on a compacted kaolin. Figure 14.14a also shows isotropic compression curves, obtained onan “artificial loess”, a silt from Trois-Rivières, Québec, reconstituted at a high void ratio of 1.04.Figure 14.15 shows compression curves obtained along stress paths with constant net stress ratio dif-ferent from 1.0 on the same Trois-Rivières silt. Figure 14.16 shows compression curves deduced fromCRS tests performed at constant suction on a residual soil (50% sand, 45% silt and 5% clay) reconsti-tuted (Figure 14.16a) and on the same soil reconstituted at the same void ratio, but with 2% cement

179

0

0.1

0.2

0.3

0.4

0.5

10 10 10

² ∆e

p - u (kPa)a2 3

e ~ 1.04initial

wau - u (kPa)

600400200150

80(a)

e ~ 1.04e ~ 0.8e ~ 0.7

ooo

(b)

800

600

400

200

00 10 20 30 40 50 60 70 80

u -

u

(kP

a)a

w

(p - u ) (kPa)a y

Elastic

LCe = 1.04o

Figure 14.14. Isotropic compression tests on Trois-Riviéres silt subjected to different suctions: (a) Compressioncurves when eo � 1.04; (b) Loading collapse curves (after Maâtouk et al., 1995).

10 10 10

p - u (kPa)a2 3

1.1

1.0

0.9

0.8

0.7

0.6

0.5

e

1.00 0.74 0.54

K = - ua3

1

σσ - ua

Figure 14.15. Compression tests at various stress ratios, for (ua � uw) � 80 kPa. Trois-Rivières silt (from Maâtouket al., 1995).

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added (Figure 14.16b). In all cases, the compression of soil under a given net stress ( n � ua) decreaseswhen suction increases. In some of the tests performed on the reconstituted residual soi (Figure 14.16),suction was decreased to zero after reaching a volumetric strain larger than 10%; it can be seen that soilconditions then move on or close to the compression curve corresponding to zero suction.

The compression curves previously presented also show that the gross yield stress increases withsuction. Alonso et al. (1990) suggested plotting this yield stress versus suction and calling the corre-sponding relationship the “loading-collapse” (LC) curve. Such curves are shown in Figures 14.14b forthe Trois-Rivières silt and 14.17 for the reconstituted residual soil non-cemented and cemented. In thislatter case, the axial yield stress is equal to 52 kPa when the soil is saturated and progressively increaseswith suction to reach a value of 175 kPa at a suction of 500 kPa. In the case of the “artificial loess” thatwas not standing on its own without suction, the LC curve does not cross the (p � ua) axis at a suctionof zero.

It has sometimes been observed that the slope of the compression curve in the e–ln(p � ua) dia-gram and in the post-yield range, s, decreases as suction increases. However, Wheeler & Sivakumar(1995 and Figure 14.13), Sharma (1998), Leroueil & Barbosa (2000, and Figure 14.16) and Machado& Vilar (2002*) found an opposite tendency. As indicated by Wheeler & Karube (1996), this could berelated to the stress range investigated.

180

10 10 100

5

10

15

20

25

30

Ver

tical

str

ain

(%)

(σ - u ) or (σ - u ) (kPa)a a a w2 3

Cemented (2%)

(u - u ) (kPa)a w0

50 100 200 300 500

35

10 10 100

5

10

15

20

25

30

Ver

tical

str

ain

(%)

(σ - u ) or (σ - u ) (kPa)a a a w2 3

CRS tests(u - u ) (kPa)a w

0 30 50

100 300 500

Non-cemented (b)(a)

Figure 14.16. One-dimensional compression curves for non-cemented and cemented (2%) Viçosa residual soil(from Leroueil & Barbosa, 2000).

500

400

300

200

100

00 40 80 120 160 200

(σ - u ) (kPa)a a

Cement0%2%

σ y, e

σ∆ σ∆y, suc y, st

∆σy, st + suc

LC (2%)LC (0%)a

a

Suct

ion

(kPa

)

Figure 14.17. Loading-Collapse (LCa) curves for cemented and non-cemented Viçosa residual soil (from Leroueil& Barbosa, 2000).

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14.4 Combined influence of matric suction and microstructure

To understand the combined influence of matric suction and microstructure on unsaturated soil behav-iour, and in particular yielding, it may be important to keep in mind the basic phenomena intervening atthe level of particles or aggregates, in particular that yielding mostly corresponds to inter-particle slip-pages, even when loading is isotropic. Figure 14.18 shows a simple contact between two particles. If thetwo particles are subjected to an external force with a component normal to the contact Fn, the resis-tance to slippage, Tn, is equal to Fntan �� for a cohesionless material (Figure 14.18a). If the same contact is bonded (Figure 14.18b), the resistance to slippage is increased by the strength of the bond, Tst.If the soil is partially saturated, there is some water surrounding the contact, a capillary force, Fsuc, normal to the contact and a slippage force, Tsuc � Fsuc tan �� (Figure 14.18c). The total resistance toslippage could simply be the sum of Tn, Tst and Tsuc. However, as indicated by Alonso et al. (1994), theresistance and thus the yield stress resulting from the combined effect of applied stress, partial satura-tion and bonding could be larger than the sum of the contributions of the three factors. This can be dueto different distributions of water at the contact between particles or, as mentioned by Alonso et al.(1994), to bond strength that is influenced by water content and indirectly suction. For the case pre-sented in Figure 14.18, this would result in an additional component of the resistance to slippage Tst�suc.The total resitance to slippage could then be written as follows:

T � Tn � Tst � Tsuc � Tst�suc (14.9)

This is for a single contact. For a soil specimen, yielding results from slippage of many contacts.However, the yield stress should be described in the same form as Equation 14.9:

y � y,e � � y,st � � y,suc � � y,st�suc (14.10)

in which y,e would be the gross yield stress associated with the void ratio; � y,st reflects the influenceof bonding alone; � y,suc reflects the influence of partial saturation alone; and � y,st�suc reflects thecombined effect of microstructure and partial saturation.

As previously indicated, Leroueil & Barbosa (2000) performed a series of CRS oedometer tests atdifferent suctions on a residual soil reconstituted and on the same soil reconstituted at the same voidratio, but with 2% cement added. The results are presented in Figures 14.16a and b respectively. Thegross yield stresses deduced from these tests are reported in Figure 14.17 as a function of suction. Inspite of some variability, the Loading-Collapse curves, called here LCa curves to denote that yielding isreached in the axial direction, are well defined. The LCa curve for the cemented material has beendrawn taking into account all the results obtained in these oedometer tests and triaxial tests that wereindicating that the yield stress obtained at a suction of 300 kPa was too small in comparison with otherresults (Leroueil & Barbosa, 2000). This figure can be used to examine the influence of void ratio,bonding and partial saturation on yielding. The yield stress y,e for the non-cemented material at zerosuction reflects the influence of void ratio; the difference between the yield stresses obtained in satu-rated conditions for the soil cemented and non-cemented, � y,st, reflects the influence of bonding alone;and the influence of suction alone is indicated by the Loading Collapse curve of the non-cementedmaterial, � y,suc. The results show, however, that, as predicted by Alonso et al. (1994) and Equation14.10, the yield stress resulting from the combined effect of void ratio, partial saturation and bonding islarger than the sum of the contributions of the three factors.

181

F

TT

F

T

u < u suc

suc

n

n st

w a

(a) (b) (c)

Figure 14.18. Resistance to slippage at a contact between particles due to: (a) external force; (b) bonding; and (c) partial saturation.

09031-02[2].qxd 18/Oct/02 12:12 AM Page 181

Carvalho et al. (2002) showed a similar combined effect of partial saturation and microstructurewhen comparing the results of isotropic compression tests performed on a silica sand cemented at dif-ferent contents of Portland cement.

14.5 Barcelona-type models for unsaturated soils

Alonso et al. (1987, 1990) extended the concepts of limit and critical states to unsaturated soils. Theproposed model, the Barcelona Basic Model (BBM), is essentially based on two considerations: first,the behaviour of unsaturated soils is controlled by the net total stress ( � ua) and the matric suction(ua � uw); second, the proposed model is an extension of the Modified Cam-clay model (Roscoe &Burland, 1968).

An important component of the approach is the Loading-Collapse (LC) yield locus and surface. Theconcept is depicted in Figure 14.19 and can be explained as follows: if a saturated soil isotropically con-solidated has an isotropic preconsolidation pressure such as po*, its limit state curve is Ysat inFigure 14.19b. If suction is increased, the isotropic yield stress increases along the LC curve as seen inSection 14.3 (Figure 14.19a) and the entire limit state curve increases in size, as shown in Figure 14.19b.It results in an enlargement of the elastic zone. In BBM, there would also be an elastic limit, the “SuctionIncrease (SI)” line corresponding to the maximum suction experienced by the soil (Figure 14.19a). In a(p � ua)–q–s diagram (Figure 14.20), the limit state curves generate a limit state surface. As shown in Figure 14.19b, the critical state line moves upwards when suction increases. In the model proposedby Alonso et al. (1990), it is considered that the slope (thus the friction angle) is constant and inde-pendent of suction.

Figures 14.12 to 14.17 can be understood by using Figure 14.19a and considering initial conditionsat a point such as I. Wetting of unsaturated specimens under constant net total stress (Figure 14.12) isassociated with a decrease in suction and collapse at Y2 (Figure 14.19a); increase in net total stressunder constant suction first shows an elastic behaviour and then development of plastic volumetricstrains after yielding at a point such as Y1 (Figure 14.19a). The volumetric strains generated in bothcases cause a movement of the LC curve towards larger net total stresses.

182

Elastic zone

s

q

p - ua

p - ua

p*

Y1

Y2

fs(s)

MIncreasingsuction

p*

o

o

(a)

(b)

satY

I

SI

Figure 14.19. Limit state curves and elastic domains for unsaturated soils – BBM (after Alonso et al., 1990).

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As it is an extension of the Cam Clay model, BBM has the same weaknesses. In particular it does notconsider the possible effects of anisotropy, viscosity and microstructure. It also considers that the grossisotropic yield stress increases as soon as there is some matric suction (Figure 14.19), and this is prob-ably not true when suction does not exceed the air entry value and the soil remains saturated.

Several other elasto-plastic models, similar to the BBM model, have been proposed to describe andpredict the behaviour of unsaturated soils: Karube (1988) and Karube & Kato (1989) proposed one inwhich the following equation is used for the critical state line:

q � M((p � ua) � f1(ua � uw)) (14.11)

This equation is similar to the one obtained by Fredlund et al. (1995, 1996) and Vanapalli et al.(1996) when they relate the shear strength and the soil-water characteristic curve (Equation 14.8); themodel proposed by Wheeler & Sivakumar (1995) adds one state parameter, the water content; Cui &Delage (1996) introduce anisotropy; Geiser et al. (1999, 2000) use ( n � uw) and (ua � uw) as stressstate variables; Buisson & Wheeler (2000) and Wheeler (2002) are working on the development of amodel that would incorporate the influence of hydraulic hysteresis.

Existence of the elastic limit SI at constant suction (Figure 14.19) has been subject to discussions, inparticular because it does not seem appropriate for unsaturated expansive clays (Gens & Alonso, 1992).These latter authors considered BBM appropriate for describing the volumetric behaviour of unsatu-rated soils of low plasticity and suggested an enhanced version, called BExM (Barcelona ExpansiveModel), for expansive clays. The BExM is clearly explained by Alonso et al. (2001) and only generalaspects are described here. Two levels are distinguished:

• The microstructural level corresponding to the arrangements of clay particles. These arrangementscan exist in a homogeneous matrix or in aggregates. It is likely that this level remains saturated andthat effective stress concepts hold. The behaviour would be essentially elastic and volumetric.

• The macroscopic level accounts for larger scale features such as aggregates. Coupling betweenmicrostructure and macrostructure may result in a build-up of macrostructural elastoplastic strainswhen elastic microstructural strains take place.

The behaviour of the macrostructure follows the BBM and is in particular characterized by the LCcurve (Figure 14.21). As microstructure is considered saturated and elastic, volumetric strains dependonly on p� and are thus equal to zero along the Neutral Line (NL at 45° in s vs (p � ua) diagram;Figure 14.21) corresponding to p� � p � uw � p � s � cst. As microstructural swelling and microstruc-tural shrinkage may affect the macrostructure and induce irreversible changes in void ratio, two addi-tional yield curves parallel to NL were introduced in BExM. These two yield curves SI (Suction Increase)and SD (Suction Decrease) are associated with shrinkage and swelling respectively (Figure 14.21).More details on BExM can be found in Gens & Alonso (1992) and Alonso et al. (1994, 2001).

183

s

q

SIp*o

p - ua

M

M

(p , s)o LC

Figure 14.20. Three-dimensional view of the yield surface in BBM (after Alonso et al., 1990).

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14.6 Limit state curves of unsaturated soils

As for saturated soils (Section 3), limit states of unsaturated soils can be defined by a variety of testsand, up to now, limit state curves have been defined for about 10 different unsaturated soils. Most ofthese soils were reconstituted and Machado & Vilar (2002*) are presenting an exceptional set of limitstate curves determined on a naturally unsaturated colluvium.

Figure 14.22 shows the limit state curves defined by Machado & Vilar (2002*) on Sao Carlos collu-vium at suctions of 0, 100 and 200 kPa. The extension of the yield curve with suction is quite clear.

184

Figure 14.21. BExM: yield loci for microstructure and macrostructure in (p � ua)–s plane (from Gens & Alonso,1992, and Alonso et al., 1994).

300

200

100

0

-1000 100 200 300 400

Saturated s = 100 kPa s = 200 kPa

400

p -u (kPa)

q (k

Pa)

a

AL

CSL (s

= 0

)

GFY

, s = 200 kPa

s = 100 kPas = 0 kPa

Figure 14.22. Yielding and GFY curves for a natural weathered colluvium soil from Brazil (data from Machodo &Vilar, 2002*; curves drawn by Leroueil & Barbosa, 2000).

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From these results, it appears that the limit state curves obtained on this soil are essentially isotropic.Limit state curves obtained by Zakaria et al. (1995) on a compacted kaolin after isotropic consolidationalso seem to be isotropic (These results are presented later in Figure 14.27). Figure 14.23 presents thelimit state curves of the compacted Jossigny silt for suctions of 200, 400, 800 and 1500 kPa (Cui &Delage, 1996). These curves, as well as those obtained by Maâtouk et al. (1995), are inclined to the hor-izontal, which probably reflects an anisotropy of particle arrangement due to the mode of preparation ofthe soil specimens. This implies that, as for saturated clays, the particle arrangement and the shape ofthe limit state curve will vary depending on the stress path followed, i.e., there will be kinematic hard-ening. This is confirmed by Romero et al. (2002).

Results obtained by Wheeler & Sivakumar (1995) on a compacted kaolin also indicate that the con-cept of critical state applies to unsaturated soils. As shown in Figure 14.24, a variety of tests performedat a suction of 200 kPa give ultimate points which are on a single critical state line in both q–(p � ua)

185

1000

800

600

400

200

00 200 400 600 800 1000

p -u (kPa)

q (k

Pa)

K

s = 1500 kPa

s = 800 kPa

s = 400 kPa

s = 200 kPa

0w

s = (u - u )a w

a

Figure 14.23. Limit state curves of the compacted Jossigny silt at various suctions (from Cui & Delage, 1996).

400

300

200

100

00 100 200 300

Mean net stress (p - u ) (kPa)

Normal compression points Critical state points

a

100 200Mean net stress (p - u ) (kPa)a

50 400

Critical state line s = 200 kPa

Critical state line s = 200 kPa

Normal compression line, s = 200 kPa

2.2

2.1

2.0

1.9

1.8

Spe

cific

vol

ume

v

Dev

iato

ric s

tres

s q

(kP

a)

(a) (b)

Figure 14.24. Test paths for constant-suction shear tests conducted as s � 200 kPa (from Wheeler & Sivakumar,1995).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 185

and v–(p � ua) diagrams. However, it appears from other data shown by Wheeler & Sivakumar (1995)that the critical state lines obtained at different suctions are not parallel in an e–log (p � ua) diagram,and tend to converge at large net pressures.

An application of CSSM to unsaturated soils is illustrated in Figure 14.25. Maâtouk et al. (1995) per-formed drained triaxial compression tests ( 3 � ua � cst) at different suctions on the unsaturated siltfrom Trois-Rivières after isotropic consolidation within the limit state curve and outside, as schema-tized in Figure 14.25a. For tests starting from post-yield conditions, plastic strains develop progres-sively and the maximum deviatoric stress and critical state conditions are reached at axial strains of25–30% (Figure 14.25c). For tests starting from pre-yield conditions, the behaviour is first elastic witha rapid increase in deviatoric stress, and plastic strains develop only when the stress path has reachedthe limit state curve; critical state conditions are obtained at axial strains of 12–15%. These simple patterns of behaviour and many others reported in the literature (Wheeler & Sivakumar, 1995; Cui &delage, 1996; Geiser et al., 2000; etc.) confirm the applicability of CSSM concepts to unsaturated soils.

14.7 A simple mechanistic model for unsaturated soils

The simple yielding model initially proposed by Larsson (1977) for saturated soft clays and discussedin Section 3.6 has been extended to unsaturated soils by Leroueil & Barbosa (2000). Starting fromLarsson’s idea that “a soft clay cannot be submitted to stresses in any direction higher than the previousstresses in that direction without yielding”, the yield curve of soft clays can be approximately definedby 4 segments, as shown in Figure 3.14. Leroueil & Barbosa (2000) added the following considerations:

– When a soil (orthotropic, with the same properties in all horizontal directions) is formed, either nat-urally or artificially, it develops a particle or aggregate arrangement so that the soil skeleton has yieldstresses in saturated conditions �yv and �yh in the vertical and horizontal directions respectively. In astress diagram, the line reflecting this induced anisotropy is called the anisotropy line (AL) and ischaracterized by KAL � �yh/ �yv.

186

q

p - u

Shear

stren

gth en

velop

e

Yield curve

a

1 2

(a)

150

100

50

00 5 10 15 20 25

ε (%)

σ -

σ

(kP

a)a

r

(b)

Type 1 behaviouru - u = 150 kPaσ - u = 10 kPa

a w

3 a

a

800

600

400

200

00 5 10 15 20 25 30 35

ε (%)a

σ -

σ

(kP

a)a

r

(c)

Type 2 behaviouru - u = 150 kPaσ - u = 200 kPa

a w

3 a

Figure 14.25. Drained shear (CID) of unsaturated Trois-Rivières silt specimens: (a) Schematic stress paths; (b)specimen consolidated in the pre-yield range; (c) specimen consolidated in post-yield range (after Maâtouk et al.,1995).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 186

– As indicated in Section 14.4 (Figure 14.18 and Equations 14.9 and 14.10), matric suction generatesa resistance to slippage at the contacts between particles (or aggregates). Its global effect should thusreflect the distribution of these contacts, and the anisotropy line (AL) should be the same as for thesame soil when saturated. This is confirmed by the studies referred to in Section 14.6, which showthat yield curves obtained at different suctions keep the same shape (see Figures 14.22 and 14.23).

– The previous statement implies that, contrary to what has often been stated in the literature, the effectof matric suction on soil behaviour is not automatically isotropic when the air phase is continuous ascapillary forces then act at the contacts, with the resulting effect reflecting the distribution of thesecontacts.

The model proposed by Leroueil & Barbosa (2000) called GFY (for Given Fabric Yield) model canbe explained with the help of Figure 14.26. If a soil in saturated conditions has a limit state curve suchas OBoAoDo in Figure 14.26a, a given suction should extend the cap BoAoDo to BsAsDs, with anincrease in axial yield stress from *ayo to *ays and an increase in radial yield stress from *ryo to *rys.The variation of the axial and radial yield stresses with suction defines two Loading-Collapse curves,LCa and LCr respectively (Figure 14.26b). However, the anisotropy line, AL, remains the same inde-pendently of suction, the ratio *rys / *ays at any suction is constant and equal to KAL.

By increasing the axial yield stress from *ayo to *ays, matric suction increases the shear strength ofthe soil in compression from point Bo to point Bs, or from Bo to B�, in Figure 14.26a. It can be easilydemonstrated that this compression strength increase is equivalent to a cohesion in compression due tosuction, cc,suc, defined as follows:

(14.12)c * *c,suc ays ayo� � �� ��

� ��( )

sin cos

sin

⋅1

187

F O

D 'E

D

D

A

BB

A

B '

C

σ*

s

so

os

o

s

s

s

aysrysayo

suc. = s

suc. = o

compression

extension

σ* σ*

s

o

Suct

ion

*aysrysayoryo σ *σ * σ σ*

LC LCr

CSL (s = 0)

CSL (s = 0)

AL

(σ +σ )/2 - ua r

σ*ryo

a

( σ −

σ)/

2a

r

(a)

(b)

a

Figure 14.26. Description of the Given-Fabric-Yield (GFY) model (from Leroueil & Barbosa, 2000).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 187

This equivalent cohesion depends only on suction and on the distribution and number of contacts. So,as long as this distribution remains the same, which should be the case in non-expansive soils for net stresses smaller than yield stresses *ays and *rys, cc,suc should be a constant, as indicated onFigure 14.26a. According to Equation 14.12, this cohesion is typically 27% of the increase in axial yieldstress due to suction.

In extension, the shear strength increase due to a given suction is associated with an increase in radialyield stress from *ryo to *rys (Figure 14.26a). This corresponds to an equivalent cohesion in extension,ce,suc, equal to:

(14.13)

As in compression, if fabric remains the same for net stresses smaller than yield stresses, the cohe-sion in extension due to suction should be constant, as shown in Figure 14.26a.

Equations 14.12 and 14.13 indicate that cohesions in compression and extension due to suction aregenerally different. The ratio ce,suc/cc,suc is equal to KAL, and takes a value of 1.0 only for soils having anisotropic fabric.

The GFY model previously described depends on the hypothesis that fabric remains the same whenthere are changes in suction or net stresses. If there is a reduction in the number of contacts between par-ticles when decreasing net stresses, the cohesion due to suction should decrease; also, if fabric is mod-ified by loading the soil outside the gross yield curve or by shearing, the effect of suction should changeaccordingly. The GFY model has been described for cohesionless materials. It can, however, be easilyextended to microstructured materials, the reference being in all cases the gross yield curve in saturatedconditions.

The GFY model has been applied to the data sets of Machado & Vilar ( 2002*), Zakaria et al. (1995)and Cui & Delage (1996). The comparisons of the experimental results with the GFY model are brieflydescribed hereunder.

– As previously indicated, Machado & Vilar (2002*) tested a natural colluvium from Brazil. The grossyield points are shown in Figure 14.22. They are perfectly fitted by the GFY model, assuming anisotropic AL, which is in agreement with previous experience with residual soils (see Leroueil &Vaughan, 1990).

– Zakaria et al. (1995) tested a kaolin, compacted and then subjected to an isotropic stress history.Figure 14.27 shows points obtained at gross yielding under suctions of 0, 100 and 200 kPa as well asthe critical state lines obtained for the same suctions. It can be seen that the proposed model fits thedata points well when assuming an isotropic AL, reflecting the stress history of the material. Thestrength appears however slightly overestimated.

c * *e,suc rys ryo� � �� ��

� ��( )

sin cos

sin

⋅1

188

300

200

100

00 100 200 300 400 500

p - u a (kPa)

q (k

Pa)

CSL (s = 0)

s = 20

0 kPa

s = 10

0 kPaCSLs indicated by

Zakaria et al.

GFY, s = 0

GFY s = 100 kPa

GFY s = 200 kPa

400

s = 20

0 kPa

s = 10

0 kPa

AL

Figure 14.27. Yielding and GFY curves for compacted kaolin after isotropic consolidation (data from Zakaria et al.,1995; curves drawn by Leroueil & Barbosa, 2000).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 188

– Maâtouk et al. (1995) and Cui & Delage (1996) tested compacted silty soils and both observed limitstate curves reflecting some anisotropy. This is demonstrated on Figure 14.23 showing the resultsobtained by Cui & Delage (1996) on the compacted Jossigny silt at suctions from 200 to 1500 kPa.Trying to apply the GFY model to Cui & Delage results, Leroueil & Barbosa (2000) concluded that:(a) KAL for this compacted soil is in the order of 0.45; (b) the model overestimates the increase instrength with suction. However, Cui & Delage (1996) were estimating yielding in shear tests per-formed in the pre-yield range well before failure, which is not in accordance with the GFY modelwhich assumes full mobilization of the capillary forces. By correcting Figure 14.23 accordingly inthe pre-yield (overconsolidated) domain, Figure 14.28 is obtained. It can be seen that the GFY modelcaptures relatively well the overall behaviour of the soil, but has a tendency to overestimate thestrength.

Figures 14.22, 14.27 and 14.28 indicate that the simple GFY model that incorporates anisotropy andthe effect of suction captures relatively well the behaviour of unsaturated soil, but could overestimatethe strength increase due to suction. Some explanations for such behaviour are presently being investi-gated at Université Laval and could justify the application of a correction factor to Equations 14.12 and14.13. In any case, it is thought that not only the strength increase due to suction can be related to theSWCC of the soil (Equation 14.8; Fredlund et al., 1996), but the entire limit state curve.

The GFY model also implies that, if the soil is compressed to stresses in excess of the limit statecurve and void ratio decreases, there could be a change in SWCC and a change in cohesion associatedwith a given suction. On the other hand, if there is a decrease in the number of interparticle contactswhen the soil is unloaded under a given suction, the suction component of cohesion should decrease.

14.8 Influence of matric suction on small strain stiffness

A comprehensive study on the effect of partial saturation on small strain stiffness has been presented byVinale et al. (2001). The material tested was the compacted Metrano silty sand. Figure 14.29 shows theLoading-Collapse curves deduced from isotropic compression tests performed on the soil compacted atthe optimum and wet (wopt + 2.5%) of optimum. The results confirm that the variation of yield stresswith matric suction depends on the fabric of the soil, which is different at the optimum and on the wetside of optimum (see Figure 14.6 and associated text).

189

1200

800

600

400

200

00 200 400 600 800 1000

p -u (kPa)

q (k

Pa)

a

800

200400

s (kPa)1500

CSL (s

= 0)

GFY, s

= 15

00 kP

a

s =

800

s =

400

s =

200

AL

1000

Figure 14.28. Yielding and GFY curves for compacted Jossigny silt (data from Cui & Delage, 1996).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 189

Figure 14.30 shows how the initial shear modulus obtained from torsional shear tests varies withmatric suction. As can be seen, the curves (as drawn by the authors) indicate an effect of suction on Go.The Go values are greater than the Go value for the saturated soil, by an amount that is about the samefor the three applied stresses at a given compaction condition. The effect of suction is however differentfor the two compaction conditions. At least as a first approximation, the small strain shear modulus ofthe unsaturated soil can be expressed as follows:

Go,unsat � Go,sat � f(ua � uw) (14.14)

This equation is logical with the model shown in Figure 14.18 and described by Equation 14.10 inwhich the effect of suction on a soil with a given fabric is additional to the effect of applied stress. Thisis also consistent with the strength that is described by the addition of a component associated withapplied stress and a component associated with suction (Equations 14.4 and 14.8). Equation 14.14 ishowever different from the corresponding equations proposed by Alonso (1998) and Vinale et al. (2001)in which Go,unsat is described as the product of Go,sat and a function of suction.

Vinale et al. (2001) stated that the mechanical response observed is in conformity with the typical S-shape of the SWCC, which is probably different at the optimum and on the wet side of optimum. Atlow suction values, the soil is almost saturated and variations in (ua � uw) are equivalent to changes inmean effective stress. At larger suctions, the increase in Go reflects capillary phenomena. It is worth not-ing that, as for the gross yield stress (LC curves shown in Figure 14.29), the effect of matric suction onGo is small at values larger than 150 kPa. Other test results presented by Vinale et al. (2001) indicate thatthe shape of the G/Go versus � curves is not significantly influenced by suction.

190

Figure 14.29. Loading-Collapse curve of the optimum and wet compacted Metramo silty sand (after Vinale et al.,2001).

Figure 14.30. Initial shear modulus versus matric suction in suction controlled torsion shear tests on Metramo siltysand: (a) optimum compacted; (b) wet compacted (after Vinale et al., 2001). Note that the curves have been redrawn,by the authors.

09031-02[2].qxd 18/Oct/02 12:12 AM Page 190

Wu et al. (1984) performed resonant column tests on compacted Glacier Way silt. Figure 14.31shows Go versus the degree of saturation for a void ratio of 0.65 and different applied stresses. Severalremarks can be made: (a) Go is essentially the same in completely saturated (Sr � 100%) and dry(Sr � 0%), (Go,dry � Go,sat); (b) the component of Go in excess of Go,dry is due to the effect of suction.It can be seen in this case that Go does not vary significantly for Sr values between 100% and 70%,increases as Sr decreases to reach a peak at a degree of saturation of about 15% and finally drops toGo,dry as Sr decreases towards 0%; (c) (Go � Go,dry) is essentially the same for the three confining pres-sures. As suction is primarily related to the degree of saturation and not to the applied stress, theseresults support Equation 14.14 and the idea that at given void ratio and fabric, the effect of suction isadditional to the effect of applied stresses.

Pintado (1993; in Alonso, 1998) reports resonant column tests performed on the “Vallformés” siltysand. Figure 14.32 shows Go as a function of suction for different confining stresses. Except for theunconfined test, the increase in Go with suction in excess of Go under zero suction is about the same,independently of the applied stress, confirming the validity of Equation 14.14.

Vinale et al. (2001) also investigated the effect of strain rate on Go measured during resonant columnand torsional shear tests performed under constant matric suction. The observations were very similar

191

Figure 14.31. Low-amplitude shear modulus versus degree of saturation for Glacier Way Silt (from Wu et al.,1984).

Figure 14.32. Maximum shear moduli as a function of confining stress and suction. Compacted soil fromVallfomes (data from Pintado (1993) presented by Alonso, 1998).

09031-02[2].qxd 18/Oct/02 12:12 AM Page 191

for both optimum and wet compacted specimens. Figure 14.33 shows the normalized shear modulusGo/Go(0.1%/min) as a function of shear strain rate for the wet of optimum compacted material under aconstant mean stress of 200 kPa. The strain rate effect seems to be independent of suction.

15 GENERALITIES ON SOIL PROPERTIES

It has been shown in Part I of this paper that all soils and soft rocks behave in a consistent manner. Theframework considered uses CSSM as a sound foundation but, for understanding the behaviour of dif-ferent types of natural soils and soft rocks, it has been necessary to build on this foundation in order totake into account the influence of anisotropy, viscosity, microstructure, discontinuities and partial satu-ration, which are not considered in CSSM. This is shown in Sections 2 to 14. Part II of this paper(Sections 15 to 20) is more oriented towards soil properties.

With its mineralogy, geological history and microstructure, a soil constitutes a specific entity. Eachparameter obtained from any test is like a photograph of this entity, and all parameters, all photographsare thus linked together. In other words, the soil properties measured at a given depth in a soil depositare not independent.

When compiling the geotechnical characteristics of eastern Canada clays, Leroueil et al. (1983a)divided soil characteristics in five different categories:

– Geology and mineralogy, which determine the nature of the considered material and its geologicalhistory.

– Properties of the remoulded or reconstituted soil.– Properties of the intact soil, which reflect the microstructure of the soil and, indirectly, its geological

history.– Properties associated with the passage from intact to destructured or remoulded conditions, which

reflect the behaviour of the soil when the initial microstructure is broken.– Hydraulic properties.

Geology and mineralogy are discussed in the Overview Paper by Locat et al. (2002*). The four otherparts are respectively in the Sections 16, 17, 18 and 19. Part II of this paper is completed by Section 20on consolidation, which combines deformation characteristics and hydraulic conductivity.

Concerning the properties of intact soils, several remarks can be made:

(1) The shape of the limit state curve depends mainly on ��nc (and indirectly on plasticity index in mostcases).

(2) Strengths measured in different types of tests are on the limit state curve, but intercept it at differentpoints, depending on the stress path followed. This means that the strengths obtained by different

192

Figure 14.33. Strain rate effect on normalized shear modulus of unsaturated Metrano siltys and compacted wet ofoptimum (after Vinale et al., 2001).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 192

tests at a given depth may differ. This is illustrated by the undrained shear strength profiles obtainedon Drammen clay using the vane, SBP and triaxial compression and extension (Figure 15.1); simi-lar results were also observed on Fucino clay (Soccodato, 2002*) and on many other sites.

(3) For a given ��nc value, the ratio between two undrained shear strengths determined by different testsor between an undrained shear strength and the preconsolidation pressure should be constant. This,in particular, implies that strength ratios should be approximately constant on a homogeneous site.However, because of (1), these ratios may vary with ��nc or Ip.

(4) What has been said in (2) and (3) also applies to deformation characteristics.

Correlations in soil mechanics are thus justified. Empirical relationships must, however, be used withextreme care: (a) the chosen parameters for the relationships may not be the most appropriate and therelationship may not be general; (b) they may not apply in types of soil in which they have not beendeveloped or checked; (c) they are approximate and should not be used for final design. On the otherhand, they can be extremely useful for checking the consistency of test results, establishing preliminarydesigns, and interpolating soil characteristic profiles on the basis of a limited number of test results.These practical aspects are developed in the Overview Paper by Hight & Leroueil (2002*).

16 PROPERTIES OF SOILS IN REMOULDED OR RECONSTITUTED CONDITIONS

16.1 Generalities

The geotechnical properties of soils in remoulded or reconstituted conditions are the physical proper-ties such as grain size distribution, particle shape, water content, plastic and liquid limits, plasticityindex, liquidity index and void index; they can also include compression and recompression indices of

193

Figure 15.1. Undrained shear strength profiles from SBP, vane and laboratory tests on Drammen clay (fromLacasse et al., 1981).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 193

the reconstituted soil as well as the critical state friction angle and the stress ratio under one-dimensional compression in the normally consolidated range, Konc. Only the properties of remoulded or reconstituted soils that characterize or can be used to assess the mechanical behaviour of natural soilsare described in this section.

The properties of soils in remoulded or reconstituted conditions are sometimes described as “intrin-sic” properties. It is, however, important to realize that these properties can be influenced by several factors and are not so “intrinsic” for the following reasons.

– The consistency limits and the compression curve depend on the fact that fine-grained soils may ormay not be dried before being reconstituted. Almeida & Marques (2002*) indicate that plasticityindex decreases by 10 to 30% with oven drying of the organic Sarapui clay. Several standards in theworld still recommend drying the soil before determining the consistency limits. As it may modifythe results, it is strongly recommended that this practice be abandoned.

– Several properties may be influenced by the conditions of sample preparation. This is true for plasticsoils as well as for cohesionless soils (See Section 5.4).

– Most properties of reconstituted active clays may be influenced by soil water chemistry (Section 13),viscosity (Sections 9 and 10) and aging effects (Section 11).

– Natural soils are often laminated even at a microscale and reconstitution involves mixing of the layers.

16.2 Liquidity index

Houston & Mitchell (1969) suggested the existence of a unique relation between liquidity index, deter-mined using the Casagrande apparatus or the fall-cone penetrometer, and remoulded shear strength ofclays. The range of values obtained was rather large, which can be explained by the fact that the liquidlimit obtained using the Casagrande apparatus is not associated with a unique value of the strength ofthe remoulded clay. On the other hand, Wroth & Wood (1978) and Leroueil et al. (1983a) have shownthat, if only values of liquid limit obtained using the fall-cone penetrometer are considered, the relationbetween IL and cur is very well defined (Figure 16.1). For values of liquidity index between 0.4 and 3.0,Leroueil et al. (1983a) proposed:

cur(kPa) � 1/(IL � 0.21)2 (16.1)

Equation 16.1 shows that cur and IL are parameters that reflect the same soil characteristic, namely,the consistency of remoulded clays.

16.3 Intrinsic compression line and void index

To characterize soil microstructure in terms of compressibility, Burland (1990) referred to the intrinsiccompression line (ICL) defined for clay reconstituted in the laboratory at a water content between wLand 1.5 wL (preferably 1.25 wL) and then consolidated under one-dimensional conditions. Burland alsodefined the void index Iv as follows:

(16.2)

where e*100 and e*1000 are the void ratios corresponding to �v � 100 kPa and �v � 1000 kPa respec-tively, on the intrinsic compression line. C*c is the intrinsic compression index. When e � e*100, Iv � 0and when e � e*1000, Iv � �1. The void index is thus defined by the two variables e*100 and C*c.

According to Burland (1990), the ICL can be represented with sufficient accuracy by the followingequation:

Iv � 2.45 � 1.285x � 0.015x3 (16.3)

where x � log �v in kPa.The ICL can be easily obtained experimentally. However, Burland (1990) showed that e*100 and C*c

could be reasonably related to the void ratio at the liquid limit, eL, as follows:

e*100 � 0.109 � 0.679eL � 0.089eL2 � 0.016eL

3 (16.4)

Ie e

e e

e e

Cv

c

��

��

�100

100 1000

100*

* *

*

*

194

09031-02[2].qxd 18/Oct/02 12:13 AM Page 194

and

C*c � 0.256eL � 0.04 (16.5)

16.4 Compression and swelling indices of non-microstructured clays

Numerous correlations between the compression index of non-microstructured clays and basic soilproperties have been proposed. Skempton (1944) and Terzaghi & Peck (1948) respectively proposed:

Cc � 0.007(wL � 10) (16.6)

and

Cc � 0.009(wL � 10) (16.7)

Nagaraj & Srinivasa Murthy (1986) established a relationship between the compression index andthe liquid limit based on considerations of physical chemistry. They came with Cc � 0.2343eL, which is very close to Equation 16.5.

Nagaraj & Srinivasa Murthy (1986) also suggested the following equation, which is also close toequations proposed by other authors:

Cc � 0.39e (16.8)

195

Figure 16.1. Relationship between liquidity index and remoulded shear strength (from Leroueil et al., 1983a).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 195

It is worth noting that Tanaka (2000) found for a variety of natural and sometimes microstructuredclays that Equation 16.7 is also valid under large applied stresses. It is also worth mentioning that Cc hasto decrease as the applied stress increases since the void ratio cannot become negative. For this reason,Butterfield (1979) suggested representing compression curves in a log �v–log e diagram.

Terzaghi et al. (1996) indicated that the swelling or recompression index Cs of clayey soils mostlydepends on the mineralogy of the clay, reflected by wL, and clay fraction, CF (see Figure 18.6). For nonmicrostructured clays, the ratio Cc/Cs is typically between 4 and 6.

16.5 Critical state friction angle

The strength of normally consolidated clays or loose cohesionless soils is characterized by the criticalstate friction angle ��cs (or ��nc for clayey soils). In clayey soils, ��cs reflects the mineralogy of the parti-cles and has often been related to the plasticity index of the soil, as shown in Figure 16.2. Although thereis significant scatter, the data show a general tendency for the friction angle to decrease as plasticityindex increases. However, as also shown on the figure, attapulgite and Mexico City clay are exceptions.They can be explained by the fibrous shape of attapulgite particles and by the fact that Mexico City claycontains glass particles and diatom shell fragments, which in both cases generate friction angles inexcess of 40°. In fact, it has to be recognized that plasticity index is not a property that reflects mechan-ical behaviour. ��cs should rather be referred to the angularity or roughness of particles. This wasdemonstrated by Tanaka (2000) and Shiwakoti et al. (2002) and is illustrated in Figure 16.3. Theseresearchers examined the effect of percentage of diatoms in clay on the friction angle. Singapore clayand kaolin were considered. Without diatoms, the friction angle was slightly below the ��cs–IP relation-ship shown in Figure 16.2, with values close to 23°, but increased to about 38°with 75% diatoms,although the change in IP was small (Figure 16.3).

The friction angle at critical state of cohesionless soils depends on the mineralogy, shape and angu-larity of the particles. It also depends on the grain size distribution of the soil. On quartz sand, Koerner(1970) observed an increase in �� from 30° to 38° as particle shape changed from sub-rounded to angu-lar. Compiling ��cs of 17 well documented sands, Bolton (1986) reported values between 30° and 37°and indicated typical values of 33° for a quartz sand and 40° for a felspathic sand. Coop & Airey(2002*) indicate a typical value of 40° for carbonate sands. Miura & Yagi (2002*) show ��cs between40° and 50°, with a tendency to decrease as the confining stress increases, for volcanic soils.

16.6 Residual strength friction angle

The residual strength is reached along a surface of discontinuity after large shear displacements. It ischaracterized by a residual friction angle, ��r, without any cohesion. The residual friction angle depends

196

Figure 16.2. Values of friction angle ��nc for clays of various compositions as reflected in plasticity index (fromTerzaghi et al., 1996).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 196

on the percentage of clay particles that can be reoriented during shearing and the ability of these parti-cles to be reoriented. As plate-shaped particles are found among clay minerals most of the time, theresidual friction angle has often been related to the clay fraction, CF, of soils.

There have been several attempts to correlate residual friction angle with other soil index propertiessuch as the plasticty index Ip (� Ac � CF) that reflects the mineralogy through the activity and the clayfraction. Several of these relationships are shown in Figure 16.4. Stark & Eid (1994) related ��r to theliquid limit, clay fraction and effective normal stress (Figure 16.5).

197

Figure 16.3. Influence of diatoms on the friction angle of clays (data from Tanaka, 2000).

Figure 16.4. Residual strength: correlations with plasticity index (after Lupini et al., 1981, and Wesley, 2002*).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 197

It is worth noting that these relationships do not directly take into account the shape of the particlesand may consequently be misleading. For example, carrying out ring shear tests on highly plastic trop-ical clays containing allophane and halloysite clay minerals, Wesley (1977) found ��r values that are 10 to 15° larger than the maximum values given in Figure 16.4 and 25 to 30° larger than the valuesdeduced from Figure 16.5. Wesley (2002*) also shows residual friction angles typically between 30 and 39°,apparently independently of the plasticity index (see Figure 16.4). Also, for soils for which the particles

198

Figure 16.5. Relationship between drained residual friction angle and liquid limit (after Stark & Eid, 1994).

Figure 16.6. Horizontal stress coefficient for normally consolidated soils (modified after Mayne & Kulhawy,1982).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 198

cannot be reoriented, for example eastern Canada clays which are mainly made up of rock flour, theresidual friction angle is not significantly lower than the critical state friction angle (Kenney, 1967).

16.7 Stress ratio under one-dimensional compression in the normally consolidated range, Konc

Numerous experimental studies have shown that the at-rest earth pressure coefficient during virgin(normally consolidated) loading Konc may be expressed using the equation proposed by Jaky (1944) andwhich can be approximated by:

Konc � 1 � sin ��cs (16.9)

Data compiled by Mayne & Kulhawy (1982) confirmed the general validity of this equation(Figure 16.6). In particular, the Konc value of 0.31 obtained by Diaz-Rodriguez et al. (1992) and Diaz-Rodriguez (2002*) on Mexico City clay is consistent with the friction angle of 43° obtained on thismaterial. However, Coop & Airey (2002*) found Konc values consistently higher (by 0.05–0.12) thanthose given by Equation 16.9 for carbonate sands.

It is worth mentioning that, testing 20 natural marine clays from different parts of the world with thesame personnel and equipment, Watabe et al. (2002) found the following equations

Konc � 1.05 � sin �� (16.10)

when �� is defined on normally consolidated samples at large strains (( �1/ �3)max), and

Konc � 0.95 � sin �� (16.11)

when �� is defined on normally consolidated samples at the peak (( �1 � �3)max).

17 MECHANICAL PROPERTIES OF INTACT SOILS

17.1 Introduction

The properties of intact soils are those which reflect the mineralogy and the grain size distribution ofthe soil, its geological history and its microstructure. They are thus existing in in situ conditions, whereand when they have not been disturbed by any stress release, loading, shearing, sampling, etc. They are of paramount importance since they generally control the soil response to any change in boundaryconditions.

Numerous tools and interpretation methods have been developed for the determination of the prop-erties of intact soils since the birth of soil mechanics, about one century ago. In this paper, however, theauthors do not aim at establishing a state-of-the-art on soil characterization. They rather will limit thepresentation to some usual tests and properties.

17.2 Soil characterization

Soil characterization is a difficult task. Numerous techniques, sometimes sophisticated, exist. There arestill, however, basic parameters that are difficult to obtain. This is particularly the case for the porosityof sands. Lunne et al. (2002a*) present, an interesting discussion on this topic. In the present section,almost only characterization with cone penetration testing will be considered.

For soils in which it can be pushed, the piezocone is probably the best tool for soil characterization.When the cone penetrometer (CPT) or piezocone (CPTU) probe is pushed down at a constant rate ofpenetration, several parameters are measured. The three main parameters are:

• The tip resistance qc; however, to get the true (generally termed the “corrected”) tip resistance qt, qchas to be corrected for pore water pressure acting just behind the cone (u2).

• The pore pressure, generally u2 measured just behind the tip and used for correcting the tip resistance.However, pore pressure can also be measured at other locations.

• The friction measured along a sleeve located behind the tip, fs.

Generally, cone penetrometers have a 10 cm2 base area, an apex angle of 60° and a surface area of thefriction sleeve of 150 cm2; the rate of penetration is normally 120 cm/min (ISSMFE, 1989). Cone pene-trometers can also be equipped with one (or several) geophone(s) allowing measurement of shear wave

199

09031-02[2].qxd 18/Oct/02 12:13 AM Page 199

velocity at depth intervals or in a continuous manner. More details on cone penetration testing can beobtained in the book by Lunne et al. (1997).

For soil characterization, Wroth (1984, 1988) suggested the use of the following normalized parameters:

• Normalized tip resistance

(17.1)

• Pore pressure ratio

(17.2)

where �u � u2 � uo, with uo, pore water pressure in the ground at the considered depth.• Normalized friction ratio

(17.3)

Several soil classification charts have been presented using cone penetration test data. Robertson(1990) proposed the classification chart shown in Figure 17.1, using Qt, Fr and Bq. With the set of

Ff

qrs

t v�

� 0

Bu

qqt v0

��

�

Qq

tt v0

v0�

�

′

200

Figure 17.1. Soil behaviour type classification chart based on normalized CPT/CPTU data (after Robertson,1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 200

parameters that can be measured at depth intervals of 1 cm or less, and the use of a classification chart,the piezocone provides detailed information on soil stratigraphy. It is however important to realize that:(a) the zones shown in classification charts are approximate; (b) in stratified deposits, the measuredparameters may not be representative of the penetrated layer; and (c) classification charts may not bevalid in soils for which they have not been checked. This seems to be the case for unsaturated colluvium(Machado & Vilar, 2002*) and residual soils (Danziger et al., 1998; Mayne & Brown, 2002*; Viana daFonseca, 2002*).

Robertson et al. (1995) and Lunne et al. (1997) also suggested the classification chart shown inFigure 17.2 and based on Qt and Go/qt.

It is seen in the following Sections that the piezocone can be used for determining design parameterssuch as friction angle of cohesionless soils, preconsolidation pressure of clayey soils, small strain shearmodulus, etc. It has also been shown that piezocone data can be related to the relative density Dr ofsands. From well-documented results of tests performed in calibration chambers on five normally con-solidated sands, Lancellotta (1983) obtained the relationship shown in Figure 17.3. As indicated byJamiolkowski et al. (1985), the correlation is not directly applicable to overconsolidated sands for whichno unique relationship exists between Dr and qc.

17.3 Preconsolidation pressure, apparent preconsolidation pressure and OCR

The preconsolidation pressure, or apparent preconsolidation pressure, is usually determined by conven-tional 24-hrs oedometer tests, and most of our practical experience is based on this test. Because theycan be automated, are rapid to perform and give continuous compression curves, CRS oedometer testsare very attractive. It must be recalled, however, that they are often performed at strain rates muchhigher than those of 5 � 10�8 to 10�7s�1 associated with the conventional 24-hrs compression curveand, consequently (Section 9.1), they may give larger preconsolidation pressure values.

The preconsolidation pressure obtained in conventional oedometer tests has often been correlatedwith the net tip resistance qt obtained from piezocone test through the factor N t, as follows:

qt � vo � N t �p (17.4)

201

Figure 17.2. Soil classification chart on normalized cone resistance and small strain shear modulus (after Lunneet al., 1997).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 201

Figure 17.4 shows the �p vs (qt � vo) relationship obtained for 31 sites from the Champlain Seabasin, Québec, with preconsolidation pressures varying from 38 to 940 kPa. The correlation is excellentwith an average N t value of 3.4, independently of the plasticity index (Demers & Leroueil, 2002).Other studies (Mayne & Holtz, 1988; Larsson & Mulabdic, 1991; Mayne, 2001; Almeida & Marques,

202

10

20

40

60

80

100

50 100 500 1000

Rel

ativ

e D

ensi

ty, D

r (%

)

Cone Tip Resistance,qc/po

qc

po�vo

(�vo/po)0.5

SandTicinoOttawaEdgarHokksundHllion mines

Dr(%) = 68 [log( )�1]

2�

2�

Figure 17.3. Correlation Relative density – Normalized tip resistance for normally consolidated sands (afterLancellotta, 1983).

Figure 17.4. Relationship between preconsolidation pressure and net tip resistance (from Demers & Leroueil,2002).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 202

2002*; De Groot & Lutenegger, 2002*) indicate similar N t values generally between 3.25 and 3.5.Mayne (2001) mentioned, however, that such N t values could underestimate the preconsolidation pres-sure in fissured geomaterials.

It can also be mentioned that numerous approaches have been proposed for estimating OCR frompiezocone test data. The following studies can be mentioned: Konrad & Law (1987); Mayne (1991);Chang et al. (2001).

17.4 Strength parameters

17.4.1 Drained strength parameters of sandy soilsThe strength of cohesionless soils is characterized by a friction angle. Numerous indirect methods havebeen proposed for estimating this parameter. Two are briefly described hereunder.

Hatanaka & Uchida (1996) established empirical correlations between the peak friction angle ofsandy soils obtained on high quality undisturbed samples, retrieved by means of in situ freezing inJapanese sites, and SPT N values. The results were for an SPT efficiency of 78%. Mayne (2001)adjusted the equation for an efficiency of 60% and obtained the equation:

�� � [15.4(N1)60]0.5 + 20° (17.5)

in which (N1)60 is the N60 value corrected to a reference stress of one atmosphere (patm):

(N1)60 � N60/( �vo/patm)0.5 (17.6)

The database used was characterized by: 0.15 mm � D50 � 7.5 mm, 1.6 � Cu � 22.3 and34 � Dr � 81%. Hatanaka & Uchida (1996) also observed that available correlations are usually tooconservative. Figure 17.5a shows the relationship between �� and (N1)60 described by Equation 17.5,together with data obtained on silty sand from Atlanta and reported by Mayne (2001). It can be seen thatthe agreement is very good.

Relationships have also been suggested for evaluating �� from piezocone tests. Figure 17.5b showsthe relationship proposed by Robertson & Campanella (1983) for unaged, uncemented quartz sands.Applying the relationship shown in Figure 17.5b to Piedmont silty sand, Mayne & Brown (2002*)obtained representative results.

17.4.2 Strength parameters of clayey soilsEffective stress strength parameters of clayey soils have been discussed in Section 3.2 (Equation 3.1) foroverconsolidated clays, Section 16.5 (Figures 16.2 and 16.3) for normally consolidated clays, and inSection 16.6 for the residual friction angle. It can be added that the mobilized friction angle also

203

Figure 17.5. Peak friction angle from in situ tests; (a) from SPT data (after Hatanaka & Uchida, 1996; from Mayne,2001); (b) from CPT data (after Robertson & Campanella, 1983; from Mayne, 2001).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 203

depends on the mode of failure, compression or extension, triaxial or plane strain and more generallyon anisotropy. Relative �� values are indicated in Table 2.1.

In undrained conditions, the strength depends on the stress path followed and thus on the type of testperformed (Figure 15.1). Figure 3.5 shows that the normalized triaxial undrained compression strengthcan be approximated by suTC/ �p � sin �nc/(1 � sin ��nc), which is confirmed by data presented byKulhawy & Mayne (1990). This relationship is shown in Figure 17.6. Figure 3.7 shows that the ratio ofthe undrained shear strengths in extension and in compression typically increases from 0.30 for non-plastic soils to about 0.78 at a plasticity index of 60%. These results have been used to define an approx-imate relationship between normalized triaxial undrained extension strength suTE/ �p and the frictionangle. This relationship is also shown in Figure 17.6. As shown by Ladd (1991), the normalizedundrained shear strength obtained in direct simple shear (DSS) test is between the compression and theextension undrained shear strengths. For normally consolidated clays, Ladd (1991) found suDSS/ �pequal to 0.225. This relationship has also been drawn in Figure 17.6. It is thought that these strengthratios established for clays could also be valid for cohesionless materials.

The vane shear strength suv is often considered as a reference for undrained shear strength and hasoften been used in correlations with other strength parameters. One of these correlations is the ratiosuv/ �p versus plasticity index. Figure 17.7 shows a relationship proposed by Bjerrum (1973), dataobtained on eastern Canada clays (Leroueil et al., 1983a) and data obtained on Japanese clays (Tanaka,2002). The relationships proposed by Bjerrum (1973) and Leroueil et al. (1983a) are slightly differentbut show the same tendency. It can be added that most non-organic clays in the world more or less fol-low these relationships. However, for organic clays (Leroueil et al., 1985c) and also for Japanese clays(Tanaka, 2002), strength ratios are generally higher and apparently not related to plasticity index (seeFigure 17.7).

Lacasse et al. (1978) and Jamiolkowski et al. (1985) suggested that field vane strength should berelated to the in situ vertical effective stress and OCR using Equation 3.1. An m value of 1.0 in thisequation corresponds to a constant su/ �p ratio. Compiling data from different authors, Tavenas &Leroueil (1987) concluded that m is close to 1.0 for most natural clays and that the ratio su/ �p does notseem to be significantly dependent on OCR.

The tip resistance in a piezocone test is related to the undrained shear strength of clays as follows:

qt � vo � Nktsu (17.7)

Nkt depends on the undrained shear strengh considered. It is generally between 10 and 20, withoutany clear tendency to vary with plasticity index, when vane shear strength is considered; it is lower, withvalues between 8 and 14, when undrained triaxial compression strength is considered (Aas et al., 1986).For the time being, Nkt has to be considered as site and shearing mode specific.

204

Figure 17.6. Approximate undrained strength ratios.

09031-02[2].qxd 18/Oct/02 12:13 AM Page 204

Figure 17.8 shows the ratio of the undrained shear strengths deduced from self-boring pressuremeterand field vane tests as a function of the overconsolidation ratio for eastern Canada clay deposits. Therelationship is quite good and shows a clear decrease of the ratio suSBP/suv when the OCR increases. Thefact that SBP generally gives higher undrained shear strength than other usual tests (Figure 15.1) hasbeen analysed by Hamouche (1995) and Leroueil et al. (2002*); it seems to be associated with the vis-cous behaviour of clays.

17.5 Deformation parameters

17.5.1 Small strain shear modulusThe small strain shear moduli can be directly measured in the laboratory or deduced from shear wavevelocity Vs as indicated by Equation 4.1. The shear wave velocity can be measured in different ways: shear

205

Figure 17.7. Variation of suv / �p with plasticity index, for eastern Canada clays (from Leroueil et al., 1983a) andJapanese clays (Tanaka, 2002).

3

2

1

00 2 4 6 8

Self

-bor

ing

pres

sure

met

er s

tren

gth

/ van

e st

reng

th r

atio

OCR

Champlain sea clays (after Hamouche, 1995) Other eastern Canada clays (after Konrad et al., 1985)

Figure 17.8. Correlation of self-boring pressuremeter/vane strength ratio with the overconsolidation ratio (fromLeroueil, 1997).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 205

waves propagating vertically, with particle motion in the horizontal direction (Vs-VH); shear waves prop-agating horizontally, with particle motion in the vertical direction (Vs-HV); shear waves propagating hor-izontally, with particle motion in the horizontal direction (Vs-HH). Vs-VH � Vs-HV is used to defineGvh � Ghv � Go. Vs-HH is used to define Ghh (Section 4.2; Equations 4.8 to 4.10).

Several geophysical methods are used for determining shear wave velocities in soil deposits: down-hole (DH) tests; cross-hole (CH) tests; seismic penetration cone (SCPT or SCPTU); Spectral Analysisof Surface Waves (SASW), (Stokoe & Santamarina, 2000). The Vs-VH and Vs-HV values deduced fromDH, CH and SCPT tests are generally very similar to the values obtained by SASW testing technique(e.g. Lo Presti et al., 2002*; Soccodato, 2002*). However, Butcher & Powell (1995) showed that theshear wave velocities measured with the different techniques can be significantly different in heavyoverconsolidated clays or layered soils.

Baldi et al. (1989) proposed a relationship between Go and qc. After testing uncemented quartz sands,Rix & Stokoe (1991) found data in the hatched area of Figure 17.9, with an average relationshipbetween Go/qc and a normalized qc value close to that proposed by Baldi et al. (1989). The correlationfound by Jamiolkowski & Lo Presti (1998) for sands and gravels of Pleistocene age from MessinaStraits is also shown in Figure 17.9. It can be seen that this latter correlation is close to the averagecurve. However, as indicated by Rix & Stokoe (1991), the Go vs qc relationship for a given soil maydepend on factors such as fines content, particle angularity, etc. that are not considered in Figure 17.9.

For clays, Tanaka et al. (1994) proposed the following relationship:

Go � 50(qt � vo) (17.8)

The Go/(qt � vo) ratios obtained by Shibuya & Tamrakar (2002*), and Leroueil et al. (2002*) arealso not far from the value of 50 given in Equation 17.8, but slightly lower. Data compiled by Mayne(2001) also indicate values lower than 50 for very soft to soft clays.

The self-boring pressuremeter test (SBPT), which minimizes soil disturbance around the proble, may be used for determining the following shear moduli: (a) Go from the initial slope of the expanioncurve; (b) Gur from small unload-reload cycles that can be performed during the expansion curve. Gur isequivalent to Geq in Figure 4.11.

206

Figure 17.9. Correlation between Go and qc for cohesionless soils (modified after Rix & Stokoe, 1991).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 206

Reliable measurement of Go with SBPT is difficult, while the measurement of Gur from smallunload-reload cycles is easier. Lo Presti et al. (2001) discuss the realization and interpretation of thispart of SBPT; they also compare Gur with Go values obtained from other tests.

17.5.2 Estimation of the in situ non-linear stress-strain curveAs indicated in Section 11.3 and in Hight & Leroueil (2002*), Go measured in laboratory is often dif-ferent, generally smaller, than that deduced in situ from seismic tests. Also, the G/Go � f(�) curve isavailable only from laboratory tests. A practical question of major importance is thus to know how arepresentative in situ stress-strain curve could be obtained. Ishihara (1996) proposed the following gen-eral equation:

(17.9)

where: (a) Gs(f) and Gs(l) are secant shear moduli at a given shear strain, �, respectively in the field andin the laboratory; (b) Go(f) and Go(l) are initial shear moduli determined respectively in the field, fromin situ seismic tests, and from laboratory tests; (c) Cr is a correction factor.

If it is assumed that the G/Go � f(�) curves have the same shape in situ and in the laboratory, then Cris taken equal to 1.0. Figure 17.10 shows such an example; if the continuous line is the laboratory line,the in situ line becomes the dashed line. Ishihara (1996) proposed Cr values that are dependent on thequality of samples and shear strain.

17.6 In situ stresses and Ko

Ko, the ratio of the horizontal and vertical effective stresses in the ground, is an important parameter asit defines the initial stress conditions for any construction project. It is also one of the most difficultparameters to determine. Several methods have been proposed for determining Ko. Most of the meth-ods are based on in situ testing but some are based on laboratory testing. Quite often, also, Ko is estab-lished on the basis of empirical correlations. Soils being inelastic and with non-linear stress-strainbehaviour (see Section 4.3 and Hight & Leroueil, 2002*), it seems most unlikely that a soil specimen,which has been subjected to total and effective stress changes due to sampling, could reflect in any reli-able way the stresses it was subjected to prior to sampling. In the following paragraphs, only some insitu measurement methods and empirical correlations are indicated.

Several methods have been proposed for determining Ko in situ: hydraulic fracturing (Bjerrum &Andersen, 1972); pushed-in spade-shaped stress cell (Massarch et al., 1975); dilatometer test (Marchetti,1980); Iowa stepped blade (Handy et al., 1982); self-boring pressuremeter (Baguelin & Jezekel, 1973;

G (f) CG (l)

G (l)G (f)s r

s

oo�

207

Figure 17.10. An approach for estimating the in situ stiffness curve on the basis of the laboratory stiffness curveand in situ Go.

09031-02[2].qxd 18/Oct/02 12:13 AM Page 207

Wroth & Hugues, 1973); shear wave velocities (Stokoe et al., 1991). Except for the last one, all thesemethods were discussed in detail by Jamiolkowski et al. (1985). Fahey (1998) stated that: “of the in situmethods, the self-boring pressuremeter test still appears to be the most accurate “direct” method ofdetermining �ho”, even if he questioned the reliability and the accuracy of this approach in stiff claysand sands.

The method using shear wave velocities is very attractive (see Fioravante et al., 1998). Ko would begiven by the following equation:

(17.10)

where Cs-HH and Cs-HV are dimensional material constants that reflect the fabric anisotropy of the soiland n is a stress parameter generally close to 0.125.

Equation 17.10 was established assuming that shear wave velocity is a power function of stresses,similar to Equation 4.10 for shear modulus. Such relationships, which imply zero shear wave velocityor shear modulus under zero stress, may be valid for soils that are not microstructured, as the reconsti-tuted sands tested by Fioravante et al. (1998). For soils that are microstructured, i.e. most natural soils,Vs or Go may not depend on applied stresses only (see Section 11.2.3), and Equation 17.10 is then notapplicable.

Empirical correlations have been proposed for the estimation of Ko. The most common is:

Ko � Konc OCR� (17.11)

where Konc is the Ko value for the soil normally consolidated, OCR is the overconsolidation ratio and � is a parameter.

Mayne & Kulhawy (1982) found good correlations with measured Ko values with � � sin ��tc. Onthe other hand, Lefebvre et al. (1991) and Hamouche et al. (1995a) found � values close to 1.0 in east-ern Canada clays. There is no clear explanation for that, but other high � values seem to exist in othersensitive clay deposits, Bothkennar and Fucino in particular (Hamouche et al., 1995a).

18 PROPERTIES ASSOCIATED WITH THE PASSAGE FROM INTACT TO DESTRUCTURED,POSSIBLY REMOULDED CONDITIONS

18.1 Generalities

It has been shown in Sections 16.2 and 16.3 that the undrained shear strength of remoulded clays andthe compression curve of reconstituted clays can respectively be characterized by the IL–cur relationshipand the intrinsic compression line (ICL). These two lines can be used as references for assessing thedegree of microstructuration of natural clayey materials.

Figure 18.1 shows the cur–IL relationship for remoulded clays (Equation 16.1 and Figure 16.1). Alsoshown on the figure is an approximate IL– �v relationship assuming a cur/ �v ratio equal to 0.3. Thesetwo lines represent the behaviour of the remoulded or completely destructured clay. If the undrainedshear strength cu and the preconsolidation pressure �p of the intact soil are plotted at the liquidity index of the soil in its natural conditions, the degree of microstructuration of the clay can be assessed. The larger the distance between intact conditions and non-microstructured ones at the natural liquidity index, the more important are the bonds between particles or aggregates.

18.2 Strength sensitivity

In undrained conditions, i.e. at constant liquidity index, the distance between log cu and log cur at the natural liquidity index is the logarithm of the sensitivity or strength sensitivity, St (Figure 18.1). As curis related to the liquidity index (Equation 16.1), it follows:

St � cu/cur � cu � (IL � 0.21)2, with cu in kPa (18.1)

Equation 18.1 means that St, cu and cur are interrelated. If two of these parameters are defined, thethird one is defined too.

KV C

V Cos-HH s-HV

s-HV s-HH

1/n

�⋅⋅

208

09031-02[2].qxd 18/Oct/02 12:13 AM Page 208

It is worth mentioning that the sensitivity determined on the basis of the fall-cone penetrometer(Equation 18.1) is generally larger than the value obtained with field vane test, probably because fullremoulding cannot be obtained with this latter test.

18.3 Void index in situ, yield stress ratio and stress sensitivity

Using the void index Iv as a normalizing parameter, it is possible to compare the sedimentation com-pression curve followed in normally consolidated clay deposit with the ICL defined in Section 16.3. Thein situ void ratio eo is then used in Equation 16.2 and the corresponding void index is termed Ivo. For anormally consolidated clay, the ratio of the vertical effective stress �vo and the vertical effective stress *ve on the ICL at Ivo reflects the microstructure of the soil developed during its deposition. For mostnormally consolidated clay deposits (but not all), Burland (1990) found that the sedimentation com-pression curves lie on the same line that he called the sedimentation compression line (SCL), (Figure 18.2).The ratio of the vertical stresses on SCL and ICL at a given void index, �vo/ *ve is often equal to 4–5.

On an Iv– �v diagram, it is also possible to draw the compression curve of a natural clayey soil asobtained in an oedometer test. An example from Sault-Ste-Marie, near Chicago, is shown inFigure 18.3. In addition to �vo/ *ve it is also possible to define the �vy/ *ve ratio (where �vy is the vertical yield stress or preconsolidation pressure), termed yield stress ratio by Burland (1990), which isa measure of the increase of resistance to compression caused by the natural microstructure. Its mean-ing is very similar to that of sensitivity expressed in terms of undrained shear strength and the yieldstress ratio has been termed stress sensitivity S by Cotecchia & Chandler (2000). For the case shownin Figure 18.3, S is equal to about 15. Evaluating strength and stress sensitivities for Louiseville clay,Leroueil et al. (2002*) found St � 20 and S � 11. In fact, it is logical to find St � S as sedimented orreconstituted clays generally show strength sensitivities larger than 1.0 when on their compression line.This is illustrated in Figure 18.4 where are shown as a function of IL the remoulded strength cur, thecompression curves obtained on the clay sedimented in water with salinity of 0.2 g/l and 35 g/l, and, atthe natural liquidity index ILo, cu and �p of the intact clay. It can be seen that, at the in situ void ratio (atILo � 2.4), St � 140 whereas S � 10 or 40, depending if the clay sedimented in 0.2 g/l or 35 g/l watersalinity.

18.4 Metastability and stiffness sensitivity

Other ways for assessing the degree of microstructure have been considered. Soga & Mitchell (1996)proposed the degree of metastability (or metastability index), defined as the difference between the liquidity index of the soil at the preconsolidation pressure and the liquidity index of the reconstitutedsoil at the same effective stress.

209

2.4

2.0

1.6

1.2

0.8

0.4

010 10 10

Ar

-1 3

u σ'vor

Liq

uidi

tyin

dex,

I L

A B

c σ'u plg St

cσ'vRemoulded clay(Eq. 16.1)

12

c (kPa) =(I - 0.21)L

ur

21 10

ur

c (kPa)

LDI

Intact clay

Figure 18.1. Structure and destructuration of clays (from Leroueil et al., 1983a).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 209

Shibuya (2000) proposed a similar metastability index based on Go. The Go � IL relationship beingdefined under one-dimensional loading for both the reconstituted and the intact soil, the metastabilityindex is the difference between the liquidity index on the undisturbed line, close to gross yield, and theliquidity index of the reconstituted soil at the same Go value.

Alternatively, the Go–IL (or Go–e) relationship established for the reconstituted and the intact soilcould be used to define a “stiffness sensitivity”, Ss. Stiffness sensitivity would be the ratio of Go valueof the intact soil close to gross yield and Go of the reconstituted soil at the same void ratio. Referring to

210

5

4

3

2

1

0

–1

–2

10 1 10 10 10 10–1 2 3 4

σ' (kPa)vo

=e

–e

*e

*–

e*

vo10

0

100

1000

xxxxx

x

A-31B-87C-18

80635846

95827154413828

646240

LLLLLL

Intrinsic compression line

Sedimentationcompressionline (SCL)

Voi

d in

dex

I

Oslo fjord 98 Älvängen S.JoaquinMilazzoBaku

ShellhavenAvonmouthDrammen

DrammenDetroit

Grangemouth

A-33

Figure 18.2. Comparison between the normalized one-dimensional compression curve obtained in laboratory onreconstituted clays and in situ conditions for many normally consolidated clays (from Burland, 1990).

Figure 18.3. Example of oedometer compression curve in Iv–log �v diagram. Sault-Ste-Marie clay (after Burland,1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 210

Figure 11.20, the stress sensitivity and the stiffness sensitivity of the Jonquière clay would be approxi-mately equal, around 1.6. For Louiseville clay, strength and stress sensitivities were found equal to 20and 11 relatively at a depth of 9.2 m (Leroueil et al., 2002*). Stiffness and stress sensitivities could alsohave been evaluated at a depth of 11.5 m (Shibuya, 2000; Leroueil et al., 2002*); their value would beapproximately 4 at that depth.

18.5 Compression and swelling indices

Many empirical relationships have been established in the past between compression index and physi-cal properties of soils, mainly liquid limit and void ratio. Some, defined for non- or slightly microstruc-tured soils, have been presented in Section 16.4. The relation Cc � 0.39e (Equation 16.8) proposed byNagaraj & Srinivasa Murthy (1986) for such soils is shown in Figure 18.5a (dashed line NS). Formicrostructured soils however, as shown in Figure 11.2 for Mexico City and Grande Baleine clays, andalso in Figure 18.3 for Sault-Ste-Marie clay, the compression index observed just after gross yielding islarger than that of the reconstituted soil. Examining data from eastern Canada and from other parts ofthe world, Leroueil et al. (1983a) came to the conclusion that Cc could be approximately defined as afunction of void ratio and strength sensitivity, as indicated in Figure 18.5a for the range of consolida-tion pressures between �p and 1.4 �p. The constant St curves shown in Figure 18.5a could probably begraduated in term of S or Ss as well.

Figure 18.5b shows the Cc–eo relationship obtained for Juturnaiba organic clays which have voidratios varying from 1.2 to 9 (Coutinho & Lacerda, 1987). In comparison with Figure 18.5a, this rela-tionship would correspond to a sensitivity of 3 to 6, which is reasonable for this type of soil. For MexicoCity clay, Mesri et al. (1975) found Cc � 1.1e (Figure 18.5b). Figure 18.5c shows relationships obtainedfor several residual soils.

Figure 18.6 shows the relationship between the swelling strain index (Css � ��v/�log �v) and CF.wLdisplayed by Terzaghi et al. (1996) for reconstituted clayey soils. It can be seen that it typically increases

211

r

10-1

1 10 10-2

Cu, u vC ,σ' (kPa)

0

0.5

1.0

1.5

2.0

2.5

3.0

I Lc

D

σ'

σ'

C I

C A

C B C B

C A

Cur

ur

ur u

u

u p

Sedimented clay

Remolded clay

Intact clay

σ'vA

vB

0.2 g/lA = 0.22c

35 g/lA = 0.4c

Destructuredclay

Figure 18.4. Undrained shear strength-liquidity index and effective stress-liquidity index relations for Grande-Baleine clay (after Locat & Lefebvre, 1982; from Leroueil et al., 1985b).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 211

212

Figure 18.5. Relations between Cc and eo: (a) clays in general (from Leroueil et al., 1983a); (b) Mexico City clays(Mesri et al., 1975) and Juturnaiba organic clays (Coutinho & Lacerda, 1987); (c) Residual soils (Leroueil &Vaughan, 1990).

Figure 18.6. Swelling strain index Css as a function of clay fraction, CF, and liquid limit wL.

09031-02[2].qxd 18/Oct/02 12:13 AM Page 212

from 0.01 for a CF.wL value of 5 to 0.3 for a CF.wL value of 1000. The swelling index of microstruc-tured soils may be much smaller, as indicated by the arrows on the figure. This is demonstrated inFigure 11.3 which shows that the swelling index of Laviano clay shale (CF � 34%; wL � 48%; car-bonate content of 19%) increases as destructuration by compression progresses. The correspondingevolution of the swelling strain index with the loading-unloading cycles is indicated on Figure 18.6; itincreases from about 0.008 for the undisturbed material (U on the figure) to 0.05 after 6 cycles of load-ing-unloading, and to 0.08 after remoulding (R on the figure). The swelling index (�e/�log �v))defined on the reconstituted soil was termed the intrinsic swelling index by Burland (1990) andexpressed by Cs*. The ratio Cs*/Cs is also an indication of the degree of microstructuration and wasdefined by Schmertmann (1969) as the swell sensitivity.

18.6 Effective strength parameters

Burland et al. (1996) extended the concepts of Hvorslev (1937) and critical state soil mechanicsdescribed in Section 2 and the concepts of intrinsic behaviour proposed by Burland (1990) to assess theinfluence of microstructure on the strength parameters of natural clays. For that purpose, they suggestednormalizing strength envelopes with respect to the vertical stress on the intrinsic compression line(ICL) at the void ratio of interest, *ve. The failure envelope for reconstituted clay can then be describedby the normalized Mohr-Coulomb equation:

(18.2)

where �* is the intrinsic Hvorslev cohesion and �*e is the Hvorslev true friction angle.Burland et al. (1996) studied four different natural stiff clays. Figure 18.7 shows the normalized

Hvorslev strength envelopes for the intact and reconstituted Vallerica clay. The influence of microstruc-ture can be estimated by the distance between the two envelopes. An interesting conclusion of Burlandet al. (1996) is that ��e value for intact clay is similar to the reconstituted value and that microstructuremostly influences effective cohesion.

18.7 Influence of leaching on the characteristics of marine clays

Due to the combined effect of glacio-isostatic rebound and change in sea level, some clay deposits thatwere sedimented in a marine environment are now above the sea level. This is particularly the case inCanada, Russia and Scandinavia. Associated with this new situation is a flow of fresh water through thesediments and progressive leaching of the marine clay. As indicated, in particular by Bjerrum (1954,1967), leaching has several effects on clay characteristics: the plasticity of the clay decreases; the undrained shear strength may slightly decrease, but at an essentially constant water content; as thecur–e and �v–e curves of the leached soil, remoulded or destructured, are at void ratios smaller than the

�

� � �

�

�

ve vee* *

tan **

213

Figure 18.7. Comparison of intact and intrinsic Hvorslev strength envelopes for Vallericca clay (after Burlandet al., 1996).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 213

corresponding curves for the clay sedimented in marine environment, it follows that the sensitivity andcompression index of the leached clay are larger than those of the unleached material. Refering toFigure 18.8, St2 and Cc2, associated with leached clay, are respectively larger than St1 and Cc1 associatedwith unleached clay.

19 HYDRAULIC PROPERTIES

19.1 Generalities

The flow rate of a liquid, q, through an area A of a porous media, due to a hydraulic gradient i, is con-trolled by Darcy’s law:

q � kiA (19.1)

where k is the hydraulic conductivity of the soil.According to Poiseuille’s law, the flow through capillaries would be characterized by a hydraulic con-

ductivity proportional to the square of the radius of the tubes, or an equivalent hydraulic radius reflect-ing the flow channels in the soil. According to Kozeny-Carman equation, the hydraulic conductivity ofa soil would be proportional to Ds

2, where Ds is a characteristic grain size, and e3/(1 � e) (see, for exam-ple, Mitchell (1993) for details). Though Poiseuille’s law and Kozeny-Carman equation are better ableto represent the flow through cohesionless soils than through clayey soils, they indicate the main factorscontrolling the flow of liquid through porous materials.

For loose and uniform sands, Hazen (1911) proposed the following equation relating the hydraulicconductivity and d10, the particle diameter corresponding to 10% passing:

k (m/s) � 0.01d10 (mm)2 (19.2)

The validity of Equation 19.2 has been confirmed by Loudon (1952). Sherard et al. (1984) andKenney et al. (1985) proposed similar relationships, but referring to d15 and d5 respectively.

214

Figure 18.8. Influence of leaching on the characteristics of marine clays.

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The hydraulic conductivity of soils thus varies tremendously from soil to soil. Figure 19.1 fromManassero (1994) provides an indication of the hydraulic conductivity of soils; it typically varies by 10orders of magnitude from gravel to clays. In fact, the hydraulic conductivity of clayey materials caneven be smaller than 10�10m/s: Mesri et al. (1975) and Diaz-Rodriguez (2002*) showed a decrease inthe hydraulic conductivity of Mexico City clay from 10�8m/s at a void ratio of 15 to 10�11 at a voidratio of 2; Chandler et al. (1990) found k values between 10�10 and 10�11m/s for London clay atBradwell; Picarelli et al. (2000; 2002*) found k values between 10�10 and 10�11m/s for Bisaccia clayshale; Neuzil (1993) estimated the hydraulic conductivity of Pierre shale (South Dakota) to be from10�13 to 10�14 m/s.

For a given soil at a given void ratio, the hydraulic conductivity depends on the size of the flow chan-nels and may thus be influenced by soil fabric. This can be illustrated by test results presented byWatabe et al. (2000). Figure 14.6 shows soil water characteristic curves for a non-plastic glacial tillcompacted wet or dry of optimum. It can be seen that the air entry pressure, which is in the order 30 kPawhen the soil is compacted on the wet side of optimum, is approximately ten times larger than when thesoil is compacted on the dry side of optimum. This indicates that the pore radii of the wet compactedsoil are typically 10 times smaller than the pore radii of the dry compacted soil (see Equation 14.1).Twelve permeability tests have been performed in saturated conditions on this till, after compactionunder different conditions. Figure 19.2 presents the measured saturated hydraulic conductivity as afunction of the compaction degree of saturation at a void ratio of 0.25. It can be seen that the hydraulicconductivity changes by a factor 100 when the compaction degree of saturation decreases from valueshigher than the value at the optimum (75%) to values that are smaller (i.e. typically the square of the ratio of pore radii of the dry compacted and wet compacted soil (102)). Viana da Fonseca (2002*)also indicates that the hydraulic conductivity of undisturbed Porto residual soil can be several orders of

215

Figure 19.1. Tentative hydraulic conductivity assignment to main classes of soil (modified from Manassero, 1994and personal communication).

Figure 19.2. Hydraulic conductivity of LG-2 till as a function of the compaction degree of saturation at a void ratioof 0.25 (from Watabe et al., 2000).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 215

magnitude larger than the hydraulic conductivity of the same soil reconstituted at the same unit weightand water content.

The hydraulic conductivity also depends on the characteristics of the permeant liquid:

(19.3)

where K is the intrinsic permeability of the soil, and �p the unit weight and � the viscosity of the permeant.The hydraulic conductivity of rock masses is mostly controlled by discontinuities. In fissured clayey

soils, the hydraulic conductivity of the mass depends if the fissures are open or not.

19.2 Vertical and horizontal hydraulic conductivities in natural conditions

19.2.1 Permeability anisotropyAnisotropy in the permeability of soils is characterized by rk � kh/kv, where kh and kv are the hydraulicconductivities in the horizontal and vertical directions respectively.

Figure 19.3 shows the horizontal and vertical hydraulic conductivities of soft clays at their naturalvoid ratio, kho and kvo. Several remarks can be made: (a) for most natural soft clays, the verticalhydraulic conductivity is between 3 � 10�10 and 5 � 10�9m/s; (b) in marine homogeneous clays, rk isgenerally smaller than 1.2; (c) rk is larger in stratified deposits or varved clay deposits but generallyseems to be lower than 10.

In stiff geologically overconsolidated clays of marine origin, the anisotropy is larger than in softmarine clays, probably due to a preferential orientation of clay particles in the horizontal direction whenthe overburden pressure was increasing. Chandler et al. (1990) found a rk value in the order of 2 inLondon clay; Cotecchia (2002*) indicates a rk value of about 3 in Pappadai clay.

Permeability anisotropy is generally small in granular soils, although this may depend on the stratig-raphy and also on the shape of the particles. Comparing horizontal and vertical hydraulic conductivitiesmeasured in pairs of undisturbed samples of gravelly soils retrieved by freezing technique, Hatanakaet al. (2001) found a rk value varying from 1.1 and 1.7. On the other hand, Jamiolkowski & Lo Presti(2002*) report rk values deduced from pumping tests in Messina sand and gravel of at least 5. This highvalue could be associated with the presence of layers with different hydraulic characteristics, but alsowith gravel grains that are mainly disk-shaped and could be preferentially oriented.

k K p�

�

�

216

10 10 1010

10

k (m/s)

Bothkennar(Leroueil et al., 1992 ; Hight et al., 2002b)

New Liskeard varned clay(Chan & Kenney, 1973)

New-Jersey varved clay(Saxena et al., 1978)

Connecticut varved clay(DeGroot & Lutenegger, 2002*)

Atchafalaya(Authors' files)

-9 -8-10

ho

k(m

/s)

vo

-10

-9

-8

2

1 -

3

4

5

1

2 -

3 -

4 -

5 -

10

10

Eastern Canada claysHong KongSwedish clays

Marine clays

Figure 19.3. Permeability anisotropy in soft marine clays (from Leroueil et al., 1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 216

For considerations on the mass hydraulic conductivity of layered soils, the reader is referred toKenney (1963).

19.2.2 Hydraulic conductivity measurementIn the laboratory, the hydraulic conductivity can be measured in flexible wall permeameter (e.g. triax-ial cell) or in rigid wall permeameter (e.g. oedometer). It can also be measured in special apparatuses(radial-flow cell (Leroueil et al., 1990); cubic cell (Chan & Kenney, 1973). The hydraulic conductivitycan be directly measured by application of a constant hydraulic head or a variable hydraulic headthrough a soil specimen. It can also be obtained from continuous Controlled Gradient (CG) or CRSoedometer tests by using the following equation (Lowe et al., 1969; Smith & Wahls, 1969):

(19.4)

where H is the current height of the specimen and ub is the excess pore pressure at the base of the spec-imen. This equation is based on the assumptions that k is constant within the specimen and that the porepressure isochrone is parabolic in shape. These assumptions are not truly representative, in particularfor stresses close to the preconsolidation pressure. However, experience indicates that reasonable k val-ues can be obtained from continuous oedometer tests provided the values near �p are disregarded(Tavenas & Leroueil, 1987). Figure 19.11 shows some e–log kv relationships obtained from CRS tests.

Perrmeability tests can be performed in situ, in boreholes, in pushed-in piezometers (including theBAT system proposed by Torstensson, 1984), or in self-boring permeameter. Tests in boreholes areunreliable mainly because of the uncertainties in the shape of the cavity through which water is injected.A pushed-in piezometer displaces and remould the soil around it, which then consolidates to a lowervoid ratio. In addition, there can be clogging of the porous element by fine clay particles. As a result,pushed-in piezometers generally underestimate the k values and are not recommended for measuringthe hydraulic conductivity of clays (Tavenas et al., 1986). The self-boring permeameter, which essen-tially eliminates the difficulties associated with pushed-in piezometers, seems to give reliable results.Figure 19.4 compares hydraulic conductivity values measured by pushed-in piezometers and the self-boring permeameter at 5 different sites. On relatively homogeneous sites such as Louiseville (seeLeroueil et al., 2002*), the ratio between the 2 values is about 2; on the other hand, in the more strati-fied St-Alban clay deposit, the ratio is larger than 10. With special tests, Tavenas et al. (1986) were alsoable to define the relative contribution to the reduction in k due to remoulding of the clay and to clog-ging of the porous element (Figure 19.4).

The influence of the geometry of the porous element on the measured hydraulic conductivity wasexamined by Diène (1989) and Bouclin (1990). These authors performed permeability tests with porouselements with different length over diameter (l/d) ratios. Typical results are shown in Figure 19.5. It canbe seen that pushed-in piezometer gives a hydraulic conductivity that regularly increases with (l/d)whereas the self-boring permeameter gives higher values than the pushed-in piezometer.

kH

t

H w

b�

�

�

∂∂ 2

217

Figure 19.4. Results of permeability tests with different probes illustrating the effects of remoulding and clogging(after Tavenas et al., 1986).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 217

Diène (1989) analysed the problem and showed numerically that the radial zone of soil influenced bypermeability test carried out in situ was largely dependent on the (l/d) ratio of the porous element. Thediameter of the influenced zone typically increases by a factor of 4 when (l/d) increases from 1 to 15.As a result, for small (l/d) ratios (possibly 1 or less), the region of soil affected by the permeability testis mostly in the region that has been disturbed by the penetration of the piezometer and has reconsoli-dated. Underestimation of the hydraulic conductivity is thus expected from pushed-in-place piezome-ters with such geometry. As (l/d) increases, the influence of the disturbed zone decreases and theobtained hydraulic conductivity becomes closer to that of the intact soil. It is thought that the effect ofthe disturbed zone becomes negligible for (l/d) values larger than about 12.

The BAT permeameter developed by Torstensson (1984) and Torstensson & Petsonk (1986) is apushed-in-place piezometer with a 30 mm diameter and 40 mm high filter tip. Considering its geome-try (l/d � 1.33), it is expected that values of k will be underestimated with this equipment. This is theconclusion that was reached at Bothkennar (Leroueil et al., 1992) and at Onsoy (Lunne et al., 2002b*)where BAT system gave hydraulic conductivities generally smaller than other permeability tests.

From Equation 20.3, it is theoretically possible to deduce k values from coefficients of consolidation.However, considering the uncertainty on this latter parameter (see section 20.2), this approach is notrecommended.

19.3 Hydraulic conductivity-void ratio relationships

The relation between hydraulic conductivity and void ratio (or porosity) has been the object of numer-ous investigations. Figure 19.6 shows a relationship for chalk. For clays, it appears that there is no general relationship between the two parameters. However, Tavenas et al. (1983a) showed that e–log kvrelationships for clays are approximately linear for most clays for vertical strains up to about 20%, i.e.in the range usually encountered in practice (Figure 19.7). These relationships can then be characterizedby the hydraulic conductivity change index Ck defined as:

Ck � �e/�log k (19.5)

Tavenas et al. (1983a) also observed that Ck is simply related to the initial void ratio, eo, of the clay(Figure 19.8). As a first approximation, they proposed:

Ck � 0.5eo (19.6)

Since 1983, Equation 19.6 has generally been confirmed for soft clays and also for London clay(Chandler et al., 1990). However, as indicated on Figure 19.8 and by Cotecchia (2002*) who found a Ckvalue of 0.29 for a natural void ratio of 0.83 for Pappadai clay, the ratio Ck/eo can be slightly smallerthan 0.5 for low void ratios.

218

Figure 19.5. Hydraulic conductivity of Saint-Esprit clay as a function of the geometry of the porous element (afterBouclin, 1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 218

Al-Tabbaa & Wood (1987) and Leroueil et al. (1990) investigated the variation of both vertical andhorizontal hydraulic conductivities with void ratio when the soil is vertically compressed. Al-Tabbaa &Wood (1987) observed on kaolin an increase in rk from 1.0 at liquidity index of 1.8 to 2.7 at liquidityindex of zero. On natural clays from Sweden and eastern Canada, which are mostly made up of rockflour, Leroueil et al. (1990) observed a smaller increase in rk as the liquidity index was decreasing(Figure 19.9). rk values obtained in natural London clay and Pappadai clay are also indicated on the fig-ure, at a liquidity index close to zero; they are between 1.4 and 2.7 for London clay and about 3 forPappadai clay.

219

Figure 19.7. Variation of the vertical hydraulic conductivity with void ratio (after Tavenas et al., 1983a).

Figure 19.6. Permeability of chalk (after Scholle, 1977; ref. by Addis, 1987).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 219

220

Figure 19.8. Relationship between the permeability change indices Ckv and Ckh and the initial void ratio (fromLeroueil et al., 1990).

Figure 19.9. Permeability anisotropy of marine clays as a function of the liquidity index (after Leroueil et al.,1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 220

Leroueil et al. (1990) also investigated the variation of rk with void ratio in varved clays loaded ver-tically. The samples were from Matagami, about 800 km north of Montreal, in an area that was coveredabout 8000 years ago by the Barlow-Ojibway Lake. The variation of the vertical hydraulic conductivitywith void ratio was studied in a triaxial cell where the specimen was compressed in steps under essen-tially one-dimensional conditions. The variation of the horizontal hydraulic conductivity with void ratiowas studied in a radial-flow permeameter. Figure 19.10 shows a typical result. Figure 19.10a shows thedetailed water-content profile of the intact specimen tested in the radial-flow cell. Figure 19.10b showsthe e–log k relationships deduced from the 2 tests. kv decreases with void ratio with a Ck value close to0.5 eo. The reduction of horizontal hydraulic conductivity with vertical compression is much smaller.This is mostly due to the fact that clayey laminae are more compressible than silty laminae, reducingvertical hydraulic conductivity more significantly than horizontal hydraulic conductivity. A practicalimplication is that anisotropy in the hydraulic conductivity of varved clays (and probably of laminatedclays in general) significantly increases when they are vertically compressed.

19.4 Other factors influencing hydraulic conductivity

When considering hydraulic conductivity, other important factors are the pore fluid chemistry andhydraulic characteristics. There are three aspects: (a) the pore fluid chemistry may modify soil fabric;(b) the pore fluid may influence the characteristics of the “electric double layer”, which may obstructthe flow channels when it increases in thickness; and (c) as indicated in Equation 19.3, the hydraulicconductivity is influenced by the viscosity of the pore fluid.

The influence of viscosity on hydraulic conductivity can be observed when permeability tests areperformed at different temperatures. Figure 19.11 shows an example where kv has been determined inan oedometer cell at different void ratios at a temperature of 20°C and in CRS tests at temperatures of10 and 50°C. The ratios between the hydraulic conductivities measured at the different temperatures atthe same void ratio are essentially constant. In particular, kv(50°C)/kv(10°C) � 2.2, which correspondsto the change in water viscosity between 10 and 50°C.

The use of distilled water in permeability tests may have an adverse effect on the measured hydraulicconductivity. Figure 19.12 which shows 3 examples of clays from Sweden and Canada subjected to aflow of distilled water, confirms the decrease in k as the salinity of the outflow water decreases and thethickness of the “electric double layer” increases. The results, however, suggest that the effect becomessignificant only when the volume of permeated distilled water exceeds about half the pore volume ofthe specimen.

221

Figure 19.10. Matagami varved clay (8.8 m): (a) water content profile; (b) variation of hydraulic conductivity withvoid ratio during vertical compression (after Leroueil et al., 1990).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 221

222

Figure 19.11. Variation of hydraulic conductivity with void ratio and temperature of Saint-Roch-de-l’Achigan clay(after Marques et al., 2002).

Figure 19.12. Effect of distilled water on the results of permeability tests on three different clays (from Tavenas &Leroueil, 1987).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 222

Figure 19.13 shows the very dramatic change in hydraulic conductivity that may be observed whenchemical fluids and in particular organic compounds are permeated through clays.

19.5 Hydraulic conductivity of unsaturated soils

The hydraulic conductivity of unsaturated soils is generally characterized by the relative hydraulic con-ductivity, kwr, defined as the ratio of the hydraulic conductivity in the unsaturated condition, kwus, to that in the saturated condition, kws. The relative conductivity is often evaluated by a power function ofthe effective degree of saturation Sre (Brooks & Corey, 1964; Fredlund & Rahardjo, 1993):

(19.7)

where Sre � (Sr � Srr)/(1 � Srr), with Srr being the residual degree of saturation (see Section 14.1 andFigure 14.1), and �k is a soil parameter.

Fredlund et al. (2001) examined approximately 300 sets of permeability data obtained on a variety ofsoils. They found �k values ranging from less than 1.0 to more than 6.0. This is partly due to difficul-ties associated with the measurement of the parameters involved, but there are also other factors.Finding that the mean value for �k is 2.37 for sands and 4.34 for clays, Fredlund et al. (2001) concludedthat “there was a definite trend towards a larger �k value for soils with higher plasticity.” The parame-ter �k can also be deduced from the soil-water characteristic curve (SWCC; see Section 14.1) of the soil(Marshall, 1958; Fredlund et al., 1994). Applying this approach to the compacted LG-2 non-plastic gla-cial till (see SWCCs in Figure 14.6), Le Bihan & Leroueil (2001) found �k values between 5 and 7, values that were also confirmed experimentally. In general, �k increases from a value of about 3 for uni-form soils to larger values in well-graded materials.

20 CHARACTERISTICS OF CONSOLIDATION

When a saturated soil mass is subjected to changes in boundary conditions, generally it has a tendencyto change volume, and as a consequence, expel or absorb water. This process takes time before the soilmass can reach a steady state. Two stages are generally considered: (a) primary consolidation duringwhich the rate of settlement or strain is controlled by the ability of the soil mass to expel water, which

kk

kSwr

wus

wsre

k� � �

223

Figure 19.13. Effect of chemistry of the permeant on the hydraulic conductivity (from Fernandez & Quigley,1985).

09031-02[2].qxd 18/Oct/02 12:13 AM Page 223

is governed by Darcy’s law and hydraulic conductivity; (b) secondary consolidation during which therate of strain is controlled only by the viscosity of the soil skeleton.

20.1 Coefficient of consolidation

Before discussing the coefficient of consolidation, its measurement and how the measured values canbe used in in situ conditions, it is of interest to recall that the basic and most general equation for one-dimensional consolidation can be written (Berry & Poskitt, 1972) as:

(20.1)

where �v � vertical strain, z � depth related to initial thickness, eo � initial void ratio, e � presentvoid ratio, k � hydraulic conductivity, and u� � excess pore-water pressure.

The assumptions required to establish Equation 20.1 are that Darcy’s law is valid and the soil is sat-urated. From Equation 20.1, it is worth noting that at a given time, the strain rate �v � ��v/�t dependsessentially on the pore pressure distribution and the vertical hydraulic conductivity at that time, andvery little on soil compressibility (through (1 � eo) and (1 � e)).

In the simple case where the applied total stress, the vertical hydraulic conductivity kv, and the defor-mation modulus M � d �v/d�v, are all constant, the equation controlling consolidation takes the well-known form:

(20.2)

with

(20.3)

where M is the deformation modulus defined as d �v/d�v in one-dimensional compression.The coefficient of consolidation cv is thus not a soil parameter as such but a function of the defor-

mation modulus M and the vertical hydraulic conductivity kv. From Equation 20.3, it can be seen thatthe coefficient of consolidation will be larger in the overconsolidated domain than in the normally con-solidated domain, where the hydraulic conductivity decreases with void ratio and the deformation mod-ulus is lower. Even in this overconsolidated domain, as the bulk modulus is generally not constant, cv isnot constant either.

When cv is constant during consolidation, the degree of consolidation expressed in terms of porepressure dissipation is equal to the degree of consolidation expressed in terms of settlement, andTerzaghi’s theory perfectly applies for this one-dimensional consolidation case.

In the special case of soil treatment with vertical drains under an embankment, deformation is stillmainly vertical but water flow is mainly horizontal. The coefficient of consolidation then combines ver-tical deformation modulus M and horizontal hydraulic conductivity kh in Equation 20.3.

Among the hypotheses of Terzaghi’s theory, those of constant hydraulic conductivity and of constantdeformation modulus are questionable. k varies with void ratio (Section 19.3); constant M means thatthe effective stress-strain relationship for the soil is linear and not affected by time or strain rate, whichis seldom the case. As a consequence, Terzaghi’s theory should generally be seen as an approximationof real behaviour.

The change in coefficient of consolidation when the soil becomes normally consolidated has conse-quences on the consolidation process when the soil is loaded from an overconsolidated state to a nor-mally consolidated state. Mesri & Rokhsar (1974) and Tavenas et al. (1979) had simulated such cases(see Figure 20.1) and showed that under this condition, a step is present in the plot of the degree of con-solidation in terms of pore pressure versus logarithm of time corresponding to the passage of the pre-consolidation pressure. On the other hand, the plot of degree of settlement versus logarithm of timeshows a typical S-shape close to the Terzaghi curve associated with the final value of the coefficient ofconsolidation, i.e. the lowest value (Figure 20.1).

ckM k e

Cvw

v o

w c�

��

� �

�

( )

.

1

0 434

∂∂

∂∂

u

tc

u

zv

��

�2

2

∂∂

∂∂

∂∂

��

�

� �

�v o

wt

e

z

k

e

u

z

1

1

224

09031-02[2].qxd 18/Oct/02 12:13 AM Page 224

20.2 Evaluation of coefficients of consolidation

20.2.1 In the laboratory using oedometer test results– Conventional incremental-loading oedometer tests. Several methods have been proposed for deter-

mining coefficient of consolidation from the consolidation curve obtained from one-stage loading:Casagrande (log t) method; Taylor (�t) method; rectangular hyperbolic fitting method (Sridharan &Sreepada Rao, 1981); etc. They are all based on the assumption that the consolidation curve perfectlyfits Terzaghi’s theory. As previously indicated, this is seldom the case. Le Bihan (2002) examined theconsequences on a typical case (curve (1) in Figure 20.2). The Casagrande (log t) method is based ont50, the time necessary for reaching a degree of consolidation of 50%. If the deduced cv value, equalto 5.0 � 10�9m2/s in this case, is used to establish the theoretical degree of consolidation-time

225

Figure 20.1. Results of CONMULT program for a stress increment spanning the preconsolidation pressure, andcomparison with Terzaghi theory (from Tavenas et al., 1979).

Figure 20.2. Consolidation, experimental (1) and theoretical (2 to 4) curves.

09031-02[2].qxd 18/Oct/02 12:13 AM Page 225

relationship, curve (2) is obtained, which coincides with the experimental curve at a degree of con-solidation of 50% but not anywhere else. The cv value deduced from Taylor’s method was found equalto 8.3 � 10�9m2/s. The corresponding theoretical degree of consolidation-time relationship is curve(3), which gives good agreement with the observed behaviour in the early stages of consolidation, butdiverges considerably afterwards. Finally, cv (calc.) has been calculated on the basis of Equation 20.3,with M equal to the average modulus of deformation over the loading step and directly measuredhydraulic conductivity. That gave a cv value equal to 1.3 � 10�8m2/s. The corresponding theoreticaldegree of consolidation-time relationship is curve (4). This trend of behaviour is typical of what isgenerally observed (Leroueil, 1985b; Olson, 1985), with:

cv (log t) � cv (��t) � cv (calc) (20.4)

The main reason for this set of differences seems to come from the effect of strain rate on the com-pression curve followed during the test. If we refer to Figure 9.6a, the effective stress-strain curve fol-lowed a-b-c during one stage-loading, with high deformation modulus between a and b (Mab) and alower one between b and c (Mbc). It is different from the curve assumed by joining the end-of-loadingpoints, a-d-c, and characterized by an average deformation modulus Madc. As previously indicated(Section 20.1, Figure 20.1), the consolidation curve, defined in term of settlement, for a soil with abi-linear compression curve is mostly associated with the coefficient of consolidation correspondingto the lowest deformation modulus. Referring again to Figure 9.6a, the consolidation curve followedduring the loading stage a to c is thus mostly associated with Mbc, with the result that the coefficientof consolidation best representing the observed behaviour (cv (log t)) is smaller than cv calculatedassuming an average deformation modulus Madc.

– Continuous loading oedometer tests. CRS and controlled Gradient oedometer tests can also be usedto determine cv as a function of strain. The hydraulic conductivity given by Equation 19.4 at a giventime is then combined with the deformation modulus M calculated at the same time over a time incre-ment �t for calculating cv from Equation 20.3. The obtained cv values are close to the cv (calc) indi-cated in the previous paragraphs and Equation 20.4.

– Laboratory vs in situ coefficients of consolidation. Coefficients of consolidation have also beendeduced from observations of settlements under embankments and compared with values deducedfrom laboratory tests, generally with the Casagrande (log t) method. Leroueil (1988) compiled suchdata for 16 sites and obtained cv (in situ)/cv (lab) ratios varying between 3 and 200. When the twoextreme high values of 200 and 70 are neglected, the average is 20. Almeida and Marques (2002*)report similar ratio values ranging between 20 and 30 for Sarapui clay deposit.

There are several reasons for explaining this discrepancy:

• The cv (lab) values considered by Leroueil (1988) had generally been determined by the Casagrande(log t) method that underestimates cv in comparison with the value calculated by the Taylor (��t)method.

• As explained in Section 19.2, the hydraulic conductivity measured in laboratory using small speci-mens may underestimate the hydraulic conductivity of the soil mass, particularly in heterogeneous orstratified deposits.

• The neglect of the two- or three-dimensional aspect of field problems and of permeability anisotropy.

In agreement with Simons (1975), the authors thus recommend for practical purposes to calculate cvon the basis of permeability tests carried out in the laboratory on large specimens, or in situ.

20.2.2 In situ, from piezocone dissipation tests and SBP holding testsIn the field, coefficient of consolidation can be determined from SBP holding tests, from piezocone dissipation tests, or from calculations based on small strain shear modulus and in situ hydraulic conductivity.

The holding test proposed by Clarke et al. (1979) consists of inflating the self-boring pressuremeterup to about 10% radial strain and then maintaining a constant deformation while measuring the decayof the pore-water pressure and the variation of cell pressure, i.e. the change in effective stress. Theanalysis of these observations provides a coefficient of consolidation.

Many approaches have been developed to model the change in pore pressure with time after piezo-cone penetration in order to obtain a coefficient of consolidation (Levadoux & Baligh, 1986; Teh &Houlsby, 1991; Burns & Mayne, 1998). Most of the proposed solutions indicate a monotonic decay ofpore-water pressure with time. However, an initial increase in pore pressure before the decrease towards

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equilibrium conditions is often observed, in particular in overconsolidated materials (see examples inBurns & Mayne (1998) and Leroueil et al. (2002*)). The solution proposed by Burns & Mayne (1998)allows the simulation of both cases, depending on the overconsolidation ratio.

A very good practical question is: How representative is the coefficient of consolidation determinedfrom in situ tests? Numerous papers have been written on this topic, and the authors do not think thatthere is a definite answer yet. As they do not have the intention of reviewing this literature, they willrather show some results that reflect their own experience.

On Louiseville site, Hamouche et al. (1995b) and Leroueil et al. (2002*) determined cv or ch in boththe laboratory and in situ. In the laboratory, cv has been determined from oedometer consolidation tests,using the Casagrande (log t) method. The values obtained in the normally consolidated range werebetween 7 � 10�9 and 1.4 � 10�8m2/s, thus typically equal to 10�8m2/s. In the overconsolidationrange, due to creep, it was generally difficult to define the end of primary consolidation. In situ, the hor-izontal coefficient of consolidation was determined from SBP holding tests and from piezocone dissi-pation tests, using the interpretation method proposed by Teh & Houlsby (1991). The results are shownin Figure 20.3. Both types of test gave essentially the same results, with ch values decreasing slightlywith depth from 2 � 10�6m2/s at a depth of 3 m to 1 � 10�6m2/s at 14 m. The coefficient of consoli-dation has also been calculated on the basis of Go obtained from cross-hole tests and ko measured bothin the laboratory and in situ, at depths between 5 and 9 m, where these two parameters do not vary sig-nificantly. The calculated values were between 4 and 7 � 10�6m2/s, depending mostly on the Poisson’sratio considered. As can be seen in Figure 20.3, these values are 3 to 5 times the values deduced fromSBP holding tests and piezocone dissipation tests.

Leroueil et al. (1995) evaluated ch with the piezocone at eight Champlain Sea clay sites where coef-ficients of consolidation were also determined by other methods. At five of these sites, an in situ valuefor the normally consolidated range was deduced from settlement observations of embankments, usingAsaoka (1978)’s method. On three of them, it was also possible to deduce an in situ value in the over-consolidated range from pore pressure observations during the early stages of construction. Figure 20.4summarizes the results. The in situ coefficients of consolidation in the normally consolidated range aretypically 10 times larger than the values deduced from oedometer test results, using the Casagrandemethod. This is consistent with observations reported elsewhere (see Section 20.2.1). The in situ coef-ficients of consolidation in the normally consolidated range are also about two orders of magnitude

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Figure 20.3. Coefficient of consolidation profiles (from Hamouche et al., 1995b; Leroueil et al., 2002*).

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smaller than the values in the overconsolidated range deduced from in situ observations. The ch valuesdeduced from piezocone dissipation tests appear to be somewhere in between the field values obtainedfor the overconsolidated and normally consolidated ranges. This could possibly be due to the fact thatpore pressure dissipation around a piezocone probe involves a volume of soil that is partly intact andoverconsolidated, and partly remoulded and normally consolidated.

The coefficient of consolidation thus remains one of the most difficult geotechnical parameters todetermine.

20.3 Hypothesis A vs Hypothesis B

One important practical question concerning consolidation is to know how to define the relevant com-pression curve for in situ conditions. As indicated by Ladd et al. (1977) and Jamiolkowski et al. (1985),there are two extreme possibilities (Figure 20.5). “Hypothesis A assumes that creep occurs only after theend of primary consolidation … ” and consequently that the stress-strain curve followed in situ is the sameas the one obtained in the laboratory at the end of primary consolidation (instant compression line inFigure 20.5a). At the end of primary consolidation, the strain in situ is thus the same as in the laboratory.“Hypothesis B assumes that some sort of “structural viscosity” is responsible for creep, that this phe-nomenon occurs during pore pressure dissipation, and therefore that the strain at the end of primary con-solidation increases” with sample thickness. The compression line followed would thus be as the dashedline in Figure 20.5a, i.e. below the compression line defined in the laboratory. The question of which oneof the two hypotheses is representative of the true behaviour has been the topic of many discussions(Mesri & Choi, 1985; Kabbaj et al., 1988; Leroueil, 1988; Mesri et al., 1994, 1995; Leroueil, 1996).

There are many indications that the effective stress-strain curve followed by the soil during primaryconsolidation is not unique but depends on strain rate thus on drainage path and loading conditions. Asa result of these indications, a number of pertinent issues can be raised:

– During secondary consolidation, there are settlements and thus hydraulic gradients and excess porepressures. How does the soil know the threshold gradient below which the soil is in secondary con-solidation?

– Hypothesis A would mean that soil behaviour is controlled by excess pore pressures and not by effec-tive stresses, which is contrary to the principles put forward by Terzaghi.

– Hypothesis A would mean that soil has one constitutive equation when there are pore pressures inexcess of a reference value and another when pore pressures are close to this reference value. Thiswould mean that an element in a soil deposit has to know this reference.

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Figure 20.4. Coefficients of consolidation (from Leroueil et al., 1995).

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– There are evidences in the laboratory that effective stress-strain curves followed by soil specimens orsub-specimens depend on the strain rate history (Figures 9.3 to 9.7).

– Most of the well documented case histories show that, in accordance with the effects of strain rate andtemperature described in Sections 9 and 10, the compression curves followed by soil sub-layers underembankments (which are at low strain rates) are below compression curves obtained in the laboratoryusing high quality samples, indicating that Hypothesis A is not valid (Kabbaj et al., 1988; Leroueil,1996, 2000; Hight & Leroueil, 2002*).

Thus in the opinion of the authors, Hypothesis A is wrong.

21 CONCLUSION

This paper is an overview of the mechanical and hydraulic behaviour and properties of natural soils andsoft rocks, from soft clay to chalk. It proposes a general framework into which most geomaterials canbe qualitatively fitted whatever their geological origin. This framework is based on the concepts of limitand critical states (CSSM) that are extremely powerful for understanding soil and soft rock behaviour.However, as CSSM does not take into account important features of natural geomaterials, such asanisotropy, viscosity, microstructure, presence of discontinuities and partial saturation, it is necessary toextend the basic concepts to incorporate the influence of these factors.

The writing of the paper has been guided by three main objectives: (a) summarize the current knowl-edge on soil behaviour; (b) try to reduce the gap between evermore specialized researchers and the large

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Figure 20.5. Consolidation of clay layers of different thickness according to hypotheses A and B.

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majority of practicing engineers who may not be aware of all the recent developments, or may not knowwhere to find information on specific aspects (e.g. small strain behaviour, localization, behaviour ofunsaturated soils, effects of principal stress rotation, etc.); (c) provide a document which, hopefully,could be useful in the teaching of advanced soil mechanics.

It is obviously impossible to summarize the mechanical and hydraulic behaviour of soils and softrocks in a short conclusion. However, key aspects of this behaviour, or the recent and apparently inter-esting developments appear to be:

1) Even though it has been demonstrated for almost 30 years, it is still not well recognized that Mohr-Coulomb failure criterion has a tendency to underestimate the strength of soils for shear modesother than compression.

2) Anisotropy plays a major role in soil behaviour as it may influence small strain soil behaviour, thelimit state curve and the consequences of stress axis rotation, in particular in undrained loading.

3) The concept of limit state, initially developed for reconstituted clays, captures well the yielding ofmost natural soils. A simple model based on the idea that the shape of the limit state curve reflectsthe distribution of contacts between soil particles or aggregates is advocated.

4) With the development of both accurate instruments for measuring strains in the pre-failure stageand of geophysical methods, the understanding of the behaviour of soils and soft rocks at smallstrains has progressed considerably in the last 20 years. In particular, it has been recognised thatdynamic and static moduli are the same when examined at the same very small strains.

5) The uniqueness of critical state or the steady state line in sands, as well as the usefulness of the stateparameter, are still the subject of debate in the profession. In particular, soil crushing is known toinfluence the critical state line. The concept of collapse and static liquefaction of loose sands hasbeen fully accepted, although there are still discussions on the characteristics of the instability line.

6) It has been demonstrated that the preparation of sandy soil specimens influences strongly theirstatic behaviour during shearing. In particular, it has been shown that moist tamping can lead tocontractant behaviour while water pluviation can lead to dilatant behaviour at the same global voidratio. A difference in hydraulic characteristics could also be seen, and all this has to be attributed todifferences in fabric.

7) The presence of clay or platey particles has a very significant effect on the behaviour of granular soils.8) The importance of microstructure on the behaviour of soils is now fully recognized. This overview

confirms that microstructure is as important as void ratio and stress history for understanding of thebehaviour of natural soils. Microstructure influences yielding of soils and modifies the small strainmoduli that are then not dependent on applied stresses alone. Consequently, destructuration decreasesstiffness and strength of soils. It is also worth mentioning that, in addition to (strength) sensitivity,a number of other approaches and parameters (stress and stiffness sensitivities) have been recentlyproposed for characterizing microstructure.

9) Conditions for localization and its consequences have been examined recently; they give an inter-esting insight into the development of post-failure deformations and its consequences on strengthmobilization.

10) The concept of brittleness, which is essential to the understanding of post-failure behaviour, hasbeen generalized to be applicable to a wide range of geomaterials and for conditions that can bedrained or undrained.

11) Viscosity, i.e. the influence of strain rate and temperature, seems to exist in most geomaterials. Theeffects are more important in clayey soils but also exist in cohesionless soils. They are more impor-tant at large strains but also exist at strains as small as 10�5. The effect of viscosity on the limit statecurve and the undrained shear strength appears to be about the same for all clays. On the other hand,the critical state and the residual friction angles are almost not affected by changes in strain rate andtemperature.

12) Pore water chemistry may influence compressibility and strength characteristics of clayey soils in asignificant manner. This should be seen as an encouragement to use in situ pore water in laboratorytesting when appropriate.

13) Understanding of the behaviour of unsaturated soils has also progressed considerably in the last twodecades. It has been shown that the concept of limit state also applies to unsaturated soils. It has alsobecome clear that the soil-water characteristic curve controls to a large extent the influence ofmatric suction on the entire limit state curve, including strength, and the hydraulic conductivity.More research is however warranted, sometimes on some apparently simple aspects. For example,the three levels of strength (peak, critical state and residual) that have been clearly established insaturated soil mechanics do not seem to be so well defined in unsaturated soil mechanics.

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There are also a number of remarks on soil properties that follow from this overview:

– The piezocone that was developed about two decades ago has become essential equipment fordetailed soil investigation in soft to stiff clays and sands. Many classifications and relationships witha variety of soil parameters have been developed.

– Care should be taken over the use of the word “intrinsic” as, often, the considered characteristics canbe influenced by the mode of preparation, pore water chemistry, viscosity, etc.

– Plasticity index is generally not an appropriate parameter for characterizing the mechanical behaviourof soils. Where possible, it seems more logical to use the critical state friction angle as a reference.

– Correlations in general certainly have to be used with care in geological environments where theyhave not been developed or checked. Types of geomaterials that do not fit classical correlations seemto be clayey soils containing diatoms and tropical, often unsaturated, soils.

One problem that can be found reading this paper is the lack of consistency in the symbols used. Infact, the symbols used in geotechnical engineering vary with time and also with the authors, and, tosome extent, this problem is in itself an overview of the present situation in our profession. However,this is not satisfactory and the authors thought that all this should be homogenized. Unfortunately, theydid not find the necessary time to do so.

ACKNOWLEDGEMENTS

The authors would first like to thank the Organising Committee of the International Workshop onCharacterisation and Engineering Properties of Natural Soils for giving them the unique opportunity toprepare this Overview Paper. They in particular thank Professor Tan Thiam Soon (chair) and Dr. PhoonKok Kwang (Secretary) for their continuous encouragement. They also thank colleagues from differentparts of the world with whom, through the years and sometimes specifically for this Workshop, theyhave exchanged ideas on various aspects of soil behaviour. They finally acknowledge the help of CédricAllenou, Mingjing Jiang, Jean Parent and Julie Turgeon in the preparation of the paper and of membersof the Organising Committee who kindly reviewed it.

LIST OF SYMBOLS

aw normalized area of waterA area; soil parameterAL anisotropy lineAP air pluviationb ( 2 � 3)/( 1 � 3)Bq pore pressure ratioc soil constantcc.suc cohesion due to suction in compressionce.suc cohesion due to suction in extensioncu undrained shear strengthcur remoulded undrained shear strengthc� effective cohesionc�e Hvorslev cohesionc�pe Hvorslev cohesion parametercv coefficient of consolidationC constant relating matric suction and radius of curvature of air-water interfaceCAD anisotropically consolidated drained triaxial testCAU anisotropically consolidated undrained triaxial testCG controlled gradient (test)CH cross hole (test)CID isotropically consolidated drained triaxial compression testCIU isotropicallly consolidated undrained triaxial compression testCPT cone penetrometerCPTU piezoconeCRS constant rate of strain oedemeter testCS critical state

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CSD constant shear drained (test)CSL critical state lineCSSM critical state soil mechanicsCVDSS constant volume direct simple shearCc compression indexC*c intrinsic compression indexCs recompression indexCss swelling strain indexCs-HV, Cs-HH dimensional material constantsCk hydraulic conductivity change indexCkx hydraulic conductivity change index in the x directionCu uniformity coefficient; undrained shear strengthC�e secondary compression indexCF clay fractiond diameterdx particle diameter corresponding to x% passingDH down hole (test)DSC directional shear cellDSS direct simple shearDr relative densityDs equivalent hydraulic radiusD50 mean particle diametere void ratioe*

100 void ratio corresponding to �v � 100 kPa on the intrinsic compression linee*

1000 void ratio corresponding to �v � 1000 kPa on the intrinsic compression linee rate of void ratio changeecs void ratio on the critical state lineeg granular void ratioemax maximum void ratioemin minimum void ratioeo initial void ratioess void ratio on the steady state line(Ev)o, (Eh)o Young’s moduli under reference stress �oEvmo vertical Young’s modulus of microstructured soil under zero stressEhmo horizontal Young’s modulus of microstructured soil under zero stressEsec secant Young’s modulusEu undrained secant modulusEx Young’s modulus in the direction xfs friction measuredFr normalized friction ratioGFY name of Leroueil & Barbosa (2000)’s model for given fabric yieldingGeq shear modulus defined in small unload-reload cycleGo small strain shear modulusGmax maximum shear modulusGs secant shear modulusGt tangent shear modulusGvh, Ghv, Ghh small strain shear moduli referring to vertical (v) and horizontal (h) axesH heightHCA hollow cylinder apparatusi hydraulic gradientICL intrinsic compression lineIB brittleness indexIGB generalized brittleness indexIL liquidity indexIp plasticity indexIR relative dilatancy indexIv void indexI1, I2, I3 first, second and third stress invariantsk hydraulic conductivity or coefficient of permeability

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kx hydraulic conductivity in the x directionko initial hydraulic conductivitykwr relative hydraulic conductivitykwus hydraulic conductivity of the unsaturated soilkws hydraulic conductivity of the saturated soilK bulk modulus; stress ratio ( �3/ �1); intrinsic permeabilityK*, KAL stress ratio representing the distribution of inter-particle or inter-aggregate contacts in soilKo coefficient of earth pressure at restKonc coefficient of earth pressure at rest of the normally consolidated soilKtan tangent bulk modulusl lengthLC loading collapse curvem stress function parameterm� strain rate parameterM slope of critical state line; deformation modulusMT moist tampingMc slope of critical state line in compressionMe slope of critical state line in extensionMce 6 sin ��e/(3 � sin ��e)Mee 6 sin ��e/(3 � sin ��e)n stress function parameterN standard penetration test N-value (blows per 30 cm)Nkt cone factor related to the undrained shear strengthN t cone factor related to the preconsolidation pressureN60 N value for a SPT efficiency of 60%(N1)60 N60 value corrected to a reference stress of one atmosphereNCL normal compression lineNSP normalized soil parameterNG soil parameter describing the change in Go with time due to agingOCR overconsolidation ratioPSC plane strain cellp mean total stressp� mean effective stressp�cs mean effective stress on the critical state line at a given void ratiop�e mean effective stress on the normal compression linep�ins mean effective stress on instability linep�o initial mean effective stressp�p stress at the intersection of the swelling line with the isotropic normal consolidation linep�r reference pressurepc crushing stress of soil particlespo* isotropic preconsolidation pressure of the saturated soilps stress describing the brittleness of soilspa or patm atmospheric pressureq deviatoric stress ( 1 � 3); flow rateqc measured piezocone tip resitanceqt corrected piezocone tip resistanceqcs deviatoric stress on the critical state lineQ soil parameterQt normalized tip resistancer radiusrk anisotropy permeability ratioR soil parameterRo overconsolidation ratio defined in isotropic conditionsRC resonant column (test)s matric suction (us � uw)su undrained shear strengthsu TC undrained shear strength in compressionsu TE undrained shear strength in extensionsu DSS undrained shear strength measured in DSS test

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SBP self-boring pressuremeterSBS state boundary surfaceSC seismic cone (test)SPT standard penetration testSSL steady state lineSWCC soil water characteristic curveSr degree of saturationSre effective degree of saturationSrf submergence degree of saturationSs stiffness sensitivitySt Sensitivity or strength sensitivityS stress sensitivityt timetp peak resistance at intermediate strain; time at the end of primary consolidationTS torsional shear (test)TTA true triaxial apparatusT temperatureTn inter-particle strength due to external stressTs surface tension of waterTst inter-particle strength due to bondingTsuc inter-particle strength due to suctionTst�suc additional inter-particle component of strength due to the combined effect of suction and

bondingu pore pressureu2 pore pressure measured just behind the tip of the piezocone probe� strain rateu� excess pore pressureua air pore pressureub pore water pressure at the base of the specimenuw water pore pressure(us*)res shear deformation of shear band necessary after peak to reach residual or steady state

conditionsv specific volume (1 � e)V volumeVo initial volumeVs shear wave velocityVs-HH shear wave velocity with shear wave propagating horizontally, with particle motion in the

horizontal directionVs-HV shear wave velocity with shear wave propagating horizontally, with particle motion in the

vertical directionw water contentwL liquid limitwp plastic limitWT water pluviationYi yield curveYsat yield curve of the saturated material� angle of contact between soil particles and air-water interface; angle of the axis of the

major principal stress to the vertical; exponent used for defining Ko�k soil parameter used for defining the hydraulic conductivity of unsaturated soils�p direction of the major principal stress to the vertical� angle of the axis of the major principal stress to the vertical��p increment of plastic shear strain��p

v increment of plastic volumetric strain�u excess pore pressure�V volume change� y.st increase in gross yield stress due to bonding� ysuc increase in gross yield stress due to suction� y.st�suc increase in gross yield stress due to the combined effect of bonding and suction�1 axial strain

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�x axial strain in the x direction�e, �

e elastic strain�h horizontal strain�p, �

p plastic strain�s shear strain�v vertical strain�vol volumetric strain�x axial strain in the direction x� strain rate�1 axial strain rate�24 h axial strain rate after 24 hours in conventional oedometer test�x strain rate in the x direction�� friction angle��cs critical state friction angle��crit critical state friction angle��e Hvorslev angle of shearing resistance��mob mobilized angle of friction��nc friction angle of the normally consolidated soil��r residual friction angle��tc friction angle in triaxial compression��te friction angle in triaxial extension�b unsaturated soil strength parameter� unit weight; shear strain�p unit weight of permeant liquid�vh, �hv, �hh shear strains referring to vertical (v) and horizontal (h) axes�w unit weight of water�1 linear threshold shear strain slope of recompression curve in e–ln p� diagram; soil parameter slope of the normal compression line in v-ln p� or e–ln p� diagram; slope of the steady

state or critical state line in e–log p� diagramc slope of the steady state or critical state line in e–log p� diagram, at stresses larger than

the crushing stresss slope of the normal compression line in e–ln (p–ua) diagram for unsaturated soils� viscosity of permeant liquid� Poisson’s ratio� volumetric water content; angle of failure surface developed in laboratory test to the

horizontal�s volumetric water content at saturation�r volumetric water content at residual conditions� mass density � effective normal stress �1 major principal effective stress; axial effective stress �2 intermediate principal effective stress �3 minor principal effective stress; radial effective stress �a axial effective stress �ac axial consolidation stress �c consolidation effective stress �o reference effective stress �p preconsolidation pressure or vertical gross yield stress in one-dimensional

compression test �p iso isotropic gross yield stress �n normal effective stress �r radial effective stress �*ve vertical effective stress on the intrinsic compression line �vm maximum vertical effective stress �vy vertical yield stress or preconsolidation pressure �x effective stress in the direction x �y gross yield stress �y.e gross yield stress associated with void ratio

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1 major principal total stress; axial total stress 2 intermediate principal total stress 3 minor principal total stress; radial stress� shear stress�f shear strength�mob mobilized shear stress�p peak strength�r residual strength�vh, �hv, �hh shear stresses referring to vertical (v) and horizontal (h) axes� state parameter (eo – ess)�N normalized state parameter�� modified state parameter� effective stress parameter� normalized volumetric water content(ua – uw) matric suction or suction(ua – uw)b air entry value or bubbling pressure( – ua) net total stress

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