GE500101

Intergranular state variables and stress±strain behaviour of silty sands

S. THEVANAYAGAM� and S. MOHAN{

Relative contributions by the coarser and ®ner grains in asilty sand to its stress±strain response are affected by theintergranular matrix structure. The nature of this contribu-tion is illustrated using an intergranular matrix phase dia-gram in terms of void ratio (e), ®nes content (FC), andintergranular and inter®ne void ratios (es and ef ). Newintergranular state parameters (øs, øf ) and (es, ef ) areintroduced as state variables to characterize silty sands; es

and ef dictate the steady-state characteristics of silty sand atlow and high ®nes contents, respectively; øs and øf re¯ectthe plastic compressibility characteristics at low and high®nes contents, respectively. Using these state variables, theanticipated stress±strain±strength behaviour of silty sand incomparison to that of the host sand is presented. Similarstress-strain behaviour is expected at the same es and initialcon®ning stress ó9c, with a few exceptions.

At a constant void ratio e, es increases while ef decreaseswith addition of ®nes; a silty sand passes through differentstates. First, at low ®nes content, es (, emax,HS, the maximumvoid ratio of the host sand, case 1), and high ef , the stress±strain behaviour is primarily governed by intergranularfriction between the coarser grains. The steady-state line(SSL) is primarily dependent on es and is fairly independentof ó9c; When compared at the same es and at the same ó9c

(or at the same ø or øs), the silty sand and the host sandshow similar (not identical) stress±strain behaviour; withaddition of ®nes, as es increases, the collapse potential in-creases and the stress±strain response becomes weaker.Second, with further addition of ®nes, when es approachesor exceeds emax,HS, the SSL is in¯uenced by es and ó9c.When es is near emax,HS (case 2), the stress±strain curves forsilty sands are similar at the same es and the same ó9c (orat the same ø or øs), but different from and stronger thanthat of the host sand. At very loose states (es . emax,HS; case3), the stress±strain curves, normalized with respect to ó9c,are similar at the same es (or at the same ø or øs). Third,with further addition of ®nes beyond a threshold value, ef

becomes suf®ciently low (case 4); the ®nes impart a signi®-cant in¯uence, while the role of intergranular (coarser-grain)friction diminishes; the silty sand is expected to behavesimilarly (not identical) to the host ®nes at the same ef andøf ; at this stage, with further addition of ®nes, the collapsepotential decreases.

KEYWORDS: laboratory tests; liquefaction; pore pressures; sands;silts; stress paths.

Le roÃle relatif des grains plus gros et des grains plus ®nsdans la reÂponse contrainte-deÂformation d'un sable silteux estaffecteÂ par la structure de la matrice intergranulaire. Lanature de ce roÃle est illustreÂe au moyen d'un diagramme deconstitution de matrice intergranulaire en termes de taux depores (e), teneur en ®nes (FC) et taux de pores intergranu-laires et inter®nes (es et ef ). Nous introduisons de nouveauxparameÁtres d'eÂtat intergranulaire (øs et øf ) et (es et ef )comme variables d'eÂtat pour caracteÂriser les sables silteux;es et ef dictent les caracteÂristiques d'eÂtat stable du sablesilteux pour des teneurs en ®nes faibles et eÂleveÂes respective-ment: øs et øf re¯eÁtent les caracteÂristiques de compressibi-liteÂ plastique pour des teneurs en ®nes faibles et eÂleveÂes,respectivement. En utilisant ces variables d'eÂtat, nous preÂ-sentons le comportement anticipeÂ contrainte-deÂformation-reÂsistance du sable silteux par rapport aÁ celui du sable hoÃte.Le comportement contrainte-deÂformation devrait eÃtre simi-laire pour le meÃme es et pour une contrainte de con®nementinitiale ó9c, aÁ part quelques exceptions.

Pour un taux de pores constant e, es augmente alors queef diminue avec l'adjonction de ®nes; un sable silteux passepar diffeÂrents eÂtats: (a) quand le contenu en ®nes est faible,es (, emax:HS, le taux de pores max. du sable hoÃte, Cas-i) etquand ef est eÂleveÂ, le comportement contrainte-deÂformationest avant tout gouverneÂ par la friction intergranulaire entreles grains les plus gros; la ligne d'eÂtat stable (SSL) deÂpendavant tout de es et n'est pas treÁs deÂpendante de ó9c; quandon compare le sable silteux et le sable hoÃte aux meÃmes es etaux meÃmes ó9c (ou aux meÃmes ø ou øs), ceux-ci montrentun comportement contrainte-deÂformation similaire (mais pasidentique); avec l'adjonction de ®nes, aÁ mesure que es aug-mente, le potentiel d'affaissement augmente et la reÂponsecontrainte-deÂformation devient plus faible; (b) quand oncontinue aÁ ajouter des ®nes, quand es approche ou deÂpasseemax:HS, la SSL est in¯uenceÂe par es et ó9c; quand es serapproche de emax:HS (Cas-ii), les courbes contrainte-deÂfor-mation pour les sables silteux sont similaires pour le meÃmees et identiques pour le meÃme ó9c (ou aux meÃmes ø ou øs),mais diffeÂrentes et plus fortes que celles du sable hoÃte; pourles eÂtats treÁs meubles (es . emax:HS; Cas-iii), les courbrescontrainte-deÂformation, normaliseÂes par rapport aÁ ó9c sontsimilaires pour le meÃme es (ou aux meÃmes ø ou øs); et (c)quand on continue aÁ ajouter des ®nes au-delaÁ de la valeurseuil, ef devient suf®samment faible (Cas-iv); les ®nes ontune in¯uence signi®cative tandis que le roÃle de la frictionintergranulaire (grains plus gros) diminue; le sable silteuxdevrait avoir un comportement similaire (mais pas iden-tique) aÁ celui des ®nes hoÃtes pour les meÃmes ef et øf ; aÁ cestade, avec l'adjonction de ®nes suppleÂmentaires, le potentield'affaissement diminue.

INTRODUCTION

In the past, most of the laboratory studies on undrained stress±strain behaviour of granular soils were con®ned to relativelyclean sands. Many natural sandy soils contain a signi®cantamount of ®nes (passing sieve No. 200, particle size less than0´074 mm). Field observations of liquefaction-related failures

indicate that sites with sandy soils containing ®nes behavedifferently from sites consisting of relatively clean sands (Seedet al., 1983; Seed, 1987; Seed & Harder, 1990). Recognizingthis, several researchers have recently begun to study the effectof ®nes on stress±strain behaviour, collapse potential, steady-state strength, and cyclic response of silty sands (Chang, 1990;Georgiannou et al., 1990, 1991a, b; Chameau & Sutterer, 1994;Finn et al., 1994; Koester, 1994; Singh, 1994; Vaid, 1994;Thevanayagam et al., 1995, 1996a, b; Zlatovic & Ishihara,1995). Efforts have also been made to develop correlations ofthe effect of ®nes on resistance to liquefaction and post-liquefaction strength based on ®eld performance data (Seed et

1

Thevanayagam, S. & Mohan, S. (2000). GeÂotechnique 50, No. 1, 1±23

Manuscript received 15 July 1997; revised manuscript accepted 27 July1999. Discussion on this paper closes 31 August 2000; for furtherdetails see p. ii.� SUNY at Buffalo.{ California Dept. of Transportation.

al., 1983; Seed, 1987; Seed & Harder, 1990; Stark & Olson,1995). The experimental studies thus far followed a comparativeapproach. The behaviour of silty sands prepared at different®nes contents has been compared with the behaviour of the hostsand. The basis for comparison is not often clear. A rationalbasis is essential to extrapolate the observed differences to othersilty sands at different ®nes contents.

Traditionally, perhaps originating from the early work on the`e±log( p9)' relation proposed by Terzaghi, void ratio e has beenchosen as one of the most important state variables to character-ize the behaviour of soils. This has partly in¯uenced the workof Roscoe and co-workers (Roscoe et al., 1958, 1963; Roscoe& Burland, 1968) in their choice of ( p9, q, e) as the statevariables in critical state soil mechanics (where p9 � meaneffective stress, and q � deviatoric stress) and the formulationof the so-called steady-state concept (Poulos, 1981). The steady-state strength (at least for a restricted mode of failure) has beencorrelated with void ratio (Fig. 1(a)). Sand specimens that areat an initial state above the steady-state line (SSL) show con-tractive behaviour (C in Figs 1(b)±(e)). Those below the SSLshow dilative behaviour (D in Figs 1(b)±(e)). If the initial stateis in the vicinity of the SSL, the stress±strain behaviour is ofteninitially contractive, followed by dilation (C±D in Figs 1(b)±(e)). Combining the effects of void ratio and con®ning stress onthe observed stress±strain response of sands, alternative statevariables have been put forward to characterize the stress±strainresponse of soils. They include the state parameter ø (Fig. 1(a))(Been & Jefferies, 1985; Jefferies & Been, 1987) and the stateindex (Ishihara, 1993). Relatively clean sand specimens with thesame state parameters show similar stress±strain behaviour.Positive values for ø signify contractive behaviour, whereasnegative values for ø signify dilative behaviour. Another con-cept that has remained dormant and has been applied onlyrecently to study comparatively the behaviour of silty sands(Kuerbis et al., 1989; Pitman et al., 1994) is the concept ofsand skeleton void ratio (Mitchell, 1993). All these conceptshave been used in comparative studies to evaluate the shearstrength and collapse potential of silty sands.

Notwithstanding their merits, application of these concepts todescribe the anticipated undrained behaviour of silty sands hasfaced problems. Observations of stress±strain behaviour haverevealed confusing behaviour. Silty sands have a smaller steady-state strength compared to clean host sands at the same voidratio (or relative density). At the same void ratio, an increase in®nes content leads to a decrease in steady state strengthfollowed by an increase in strength beyond a certain limiting®nes content. This transition ®nes content has been observed tobe in the vicinity of 20±30% by weight of total solids (Pitmanet al., 1994; Zlatovic & Ishihara, 1995; Thevanayagam et al.,1996a, b). The stress±strain behaviour of different silty sandsprepared at different ®nes content, when compared in terms ofthe state parameter ø, did not always exhibit the expected trendof increased contractiveness with increase in ø as observed forclean sands (Pitman et al., 1994). Similarly, when compared interms of sand skeleton void ratio, specimens of Ottawa sandprepared at four different ®nes contents (10%, 20%, 30% and40% crushed silica ®nes), each at a progressively higher sandskeleton void ratio, showed an increased degree of dilation(Pitman et al., 1994).

While there is a better understanding of the anticipatedbehaviour of a clean sand under undrained monotonic andcyclic loading in terms of its initial state of stress, void ratioand state parameter, there is mixed opinion in the literature onthe role of ®nes in the stress±strain behaviour of a silty sand(e.g. Pitman et al., 1994; Vaid, 1994; Zlatovic & Ishihara,1995). The basis for determination of whether a silty sandwould be contractive or dilative and what kind of stress±strainbehaviour is to be expected compared with that of clean sandis less clear. What controls the behaviour of silty sandsremains to be resolved. The question is one of paramountimportance. Natural sands contain a signi®cant and varyingamount of ®nes, whereas the current knowledge is primarilybased on clean sands. As was recently pointed out

(Thevanayagam et al., 1996a, b), many of the case histories ofpost-lique-faction failure (Seed & Harder, 1990) involved soilscontaining ®nes.

The present study focuses on developing a framework forunderstanding the matrix effects and identifying new state vari-ables for characterizing the undrained triaxial stress±strain beha-viour of (gap-graded) silty sands compared with that of the hostsand and the host silt. First, the rationale for the traditional useof global void ratio (e) and state parameter (ø) as state variablesto describe the stress±strain behaviour of sands is explored. Thecritical state soil mechanics framework is extended to silty sandsmodi®ed in terms of intergranular void ratios (es, ef ) and inter-granular state parameters (øs, øf ). The stress±strain behaviourof silty sand in comparison with that of the host sand and thehost silt is presented. Although the experimental study is limitedto triaxial compression only, the concept applies to other stresspaths and modes of deformation as well.

PRESENT STUDYÐFRAMEWORK

Critical state index and plastic compressibilityThe soil microstructure is complex. It consists of a collection

of interacting clusters ordered in some organized fashion atdifferent scale levels, each dictated by different force ®elds(electrical, chemical, mechanical). The mechanisms affectingthe soil response at each scale level are not necessarily thesame, but they all collectively in¯uence the macro-behaviour.The macro-behaviour can be predicted by studying and integrat-ing the behaviour at each scale level. This concept has recentlybeen successfully applied to study ¯ow through clayey soils anddispersion phenomena in porous media (Thevanayagam, 1997;Thevanayagam & Nesarajah, 1998). It applies equally to themechanical behaviour. Evidence for this can be drawn from theexperiments on stress±strain behaviour of two-dimensional discassemblies conducted by Drescher & De Josselin de Jong(1972). The system was found to behave as an assemblyconsisting of block-like sub-regions.

When an external force acts on a soil specimen, this force isdistributed and carried internally by a hierarchy of clusters.These clusters constitute an internal force chain formed byactive grain contacts that transfer the normal forces and sustainshear forces. Active sliding, rolling and interactions occur at thecontact boundaries between the skeletons at each scale level.Active frictional contacts and mineral surface friction are theprimary mechanisms of resistance to deformation. As the ex-ternal load changes, the sub-matrices at each scale level deform.The active contacts change and reorient until an ultimate stateis reached where any further deformation does not alter thenature of active contacts at the same volume (ideally). Thistendency for change (hereafter referred to as plastic compressi-bility) dictates the macroscopic stress±strain response character-istics. The plastic compressibility itself changes as themicrostructure is altered. A set of indices that represent theactive frictional contacts and the tendency for change in activecontacts (compressibility of the sub-matrices) is needed toqualitatively characterize the stress±strain behaviour. Beforesuch indices are proposed, it is prudent to revisit the conceptsof critical state soil mechanics and identify the physical mean-ing of the state variables utilized therein in this framework.

Void ratioÐcritical state index. Reverting to a rather simplermodel, ®rst consider a perfectly arranged (body-centred cubiclattice) system of spherical particles of equal size. When actedupon by an external force, all particles may participate in theforce chain transferring and sustaining normal and shear forces.If one considers a perfect face-centred cubic lattice instead, theforce chain will be different and so will the void ratio. Hencevoid ratio could be used as an index of active contacts.

A direct extension of this analogy to sands is not perfect.Sandy soils contain particles of a wide range of size and shapearranged in a heterogeneous form. There is reason to believethat, once acted upon by an external force, some of the grains,by virtue of their size, heterogeneous positions, and availability

2 THEVANAYAGAM AND MOHAN

of void space to adjust their positions without signi®cantlyaffecting the adjacent particles, may not actively participate intransferring the normal forces or sustain signi®cant shear forces.Only a fraction of the particles may actively participate insustaining the shear forces along their contacts and form theforce chain, while the others remain relatively inactive. Voidratio is only an approximate, imperfect index of active contactsin the force chain. Rather, the void space distribution (or poresize distribution) becomes a better descriptor of active contacts(Altschaef¯ & Thevanayagam, 1991; Thevanayagam, 1989).

Nevertheless, for many practical applications, one is often leftwith void ratio as the sole index.

The question is, at what state is void ratio an index of activecontact? Limiting the discussion to uniform sands, taking un-drained triaxial compression as an example, however, the activecontacts cannot remain the same throughout the entire un-drained deformation process of a specimen (at the same voidratio). If there exists a unique ultimate steady state, in a morerestricted triaxial compression mode, then the void ratio is anapproximate index of active contacts at the ultimate steady state

Fig. 1. Undrained stress±strain behaviourÐschematic diagram

e

(5) (4)

Steady-state line(SSL)

5 Initial state(1), (2), .... 5 Specimen number

Isotropic compression line

(1)

(2)

(3)

ψ 5 State parameter for (1) (1 ve)

log p ′(a)

(1)

(2)

(3)

D

C–D

C

q

p ′(b)

C

C–D

D(5)

(4)

(1)

q

p ′(c)

C–D

D

C (1)

(2)

(3)q

εa

(d)

C–D C

D(5)

(4)

(1)

q

εa

(e)

C 5 contractionC–D 5 initial contraction followed by dilationD 5 dilation

STRESS±STRAIN BEHAVIOUR OF SILTY SANDS 3

only. The parameters ( pf , qf , e) refer to the ultimate state wherefurther shear strains do not alter the force chain or the activecontacts or cause any further plastic volumetric compression.Void ratio is an index of the microstructure of active particlesat critical state.

State parameterÐplastic compressibility index. At any otherstate, the difference between the current state ( p9, q, e) and theultimate state ( p9f , qf , e) signi®es the anticipated changes in theactive contacts (plastic compressibility) and the associateddeformation characteristics of the sand. This difference in statesexpressed in any form together with void ratio foretells theanticipated behaviour of a soil and the ultimate state. In criticalstate soil mechanics, using the so-called work-assumptionpostulate (Roscoe & Burland, 1968), the incremental plasticvolumetric compressibility (hence the contractive or dilativebehaviour) upon shearing is expressed in terms of the state ofstress (ç � q=p9) and the ultimate (critical) state (M � qf= p9f ):

Äåpv

Äåpq

� M2 ÿ ç2

2ç; where M � 6 sinö

3ÿ sinö(1a)

where Äåpv � incremental plastic volumetric strain, Äåp

q � incre-mental plastic shear strain, and ö � mobilized angle of shearingresistance at critical state. Variations of this formulation incor-porating the effects of initial and induced anisotropy can befound elsewhere (Dafalias, 1987, Thevanayagam, 1989;Thevanayagam & Chameau, 1992).

Although not widely recognized in the published literature,the above work assumption can be rewritten simply in terms ofthe critical state parameters and the state parameter (ø).

Äåpv

Äåpq

� M 2ÿ R(1(ø=ø0))ÿ �

2 R(1(ø=ø0)) ÿ 1ÿ �1=2

� f (ø, ø0, M , R) (1b)

where ø � state parameter at any stage during loading intriaxial compression, ø0 � the initial state parameter at theisotropic normally compressed state (Thevanayagam, 1989), andR � 2 according to the work assumption of Roscoe & Burland(1968). Hence, within the framework of critical state soil mech-anics, the state parameter ø has the connotation of a partialmeasure of plastic volumetric compressibility of the soil,whereas void ratio refers to its ultimate state.

While the `work assumption' may not accurately describe theinternal mechanisms of deformation, using this only as a guide,the parameters (ø, e) are expected to partially foretell theanticipated deformation characteristics and the ultimate state ofthe soil. Whether it can be used to foretell comparatively thebehaviour of silty sands at different ®nes contents based on theknowledge gained from studies on clean sands is explored next.

Critical state index and plastic compressibilityÐsilty soilsFor silty sands, the direct use of global void ratio and state

parameter as a set of unique state variables is expected to breakdown further. Due to the existence of large intergranular poresof tens of micrometers in size (between the sand grains), thereis reason to believe that, during deformation, the micrometre tosubmicrometre particles (®nes) would tend to adjust their posi-tions, moving through the pores without signi®cantly affectingthe adjacent particles. Very little opportunity exists for the ®nergrains to actively participate in transferring the normal forces orsustain signi®cant shear forces, if they are contained within theintergranular void spaces, and their quantity is smaller than athreshold value. Only a fraction of the particles may activelyparticipate in sustaining the shear forces. An analysis of theuniform-sized spherical particle system considered before butcontaining a small percentage of smaller-sized particles con-tained in it shows that the minimum size disparity for this tooccur is only about 6´5, although practical considerations woulddemand a much higher size ratio. On the other hand, noticeableparticipation is possible if the ®nes fall between the coarser

grain contacts or they are in suf®cient quantity to ®ll theintergranular space and begin to displace the coarser graincontacts. Hence, the direct use of critical state mechanicsconcepts and the associated state variables (e, ø) as uniqueindices of active contacts and plastic compressibility of siltysoils is dif®cult. They may not discriminate the differences inthe microstructure and therefore may not be directly used toforecast what to expect of a silty soil at different ®nes contents.

The problem could be simpli®ed by considering the siltysand as a delicate matrix consisting of two sub-matrices,namely, the coarser-grained matrix and the ®ner-grained matrix,and approximately analysing how they interact with each other.Consideration is given to the differences in compressibility ofthe respective individual matrices. For this purpose, in theremainder of this paper, the ®ner grains are de®ned as thosewhose size is signi®cantly less than that of the voids that wouldbe present in the soil, if it were to contain only the coarsergrains at the same void ratio. In the case of a silty sand orclayey sand, the ®ner grain size is conveniently chosen as thatpassing sieve number 200 (0´074 mm), although logically thechoice should be made on the basis of the pore size distributionof the coarser-grained matrix. In the case of a sandy gravel,sand sizes (or less) may be considered to be the ®ner grains aslong as the size disparity is large.

Intergranular void ratioÐcritical state index. At low ®nescontent less than a threshold value, it is assumed that the ®nergrains do not actively participate in the transfer of contactfrictional forces, or their contribution is secondary. The rationaleis that at such low ®nes contents the ®ner-grains matrix is highlycompressible unless the ®nes are located at the contact pointsbetween the coarser grains or when they are trapped in theintergranular void spaces at a compacted state. With thisassumption, as a ®rst-order approximation, the volume of ®nesis considered to be a part of the voids between the coarsergrains. Neglecting the differences in speci®c gravity of the ®nerand coarser grains in a silty sand, for a unit volume of solids(without the ¯uid/gas phases) containing both sizes of grains at a®nes content FC (as a percentage of the total weight of thesolids), the volume of ®nes would be FC=100 and the volume ofcoarser grains would be (1ÿ FC=100). Accordingly, the inter-granular (coarser-grains) void ratio es is de®ned as the volume ofthe voids plus the ®ner grains per unit volume of the coarsergrains:

es � [e� (FC=100)]

[1ÿ (FC=100)](2)

where e is the global void ratio of the silty sand.

Intergranular steady-state line. By analogy, the intergranularvoid ratio, es, is considered to be an index of active coarsergranular frictional contacts that sustain the normal and shearforces at the ultimate state. Then, for the same host sand, aunique intergranular steady-state line (ISSL, the locus of steady-state points in es versus log( p9f ) plane; Fig. 2(a)) is expected forsilty sands at all ®nes contents (until a threshold is reached). TheISSL is expected to coincide with the SSL for the host sand,except for the cases where the ®ner grains may also participatein the force chain, as discussed before. The traditional SSLs (eversus log( p9)) are expected to be different at different ®nescontents. Equation (2) indicates that each SSL is expected to lieprogressively lower in the e versus log( p9) plane with an increasein ®nes content (up to the threshold). A traditional SSL for asilty sand at a particular ®nes content FC � FC1 is shown in Fig.2(a). The SSL lines are expected to be, not necessarily butpractically, nearly parallel to the ISSL.

Intergranular state parameterÐplastic compressibility index. Ifindeed the intergranular contact is the primary mechanismcontributing to the stress±strain behaviour, then the compressi-bility of the material must also be attributed to intergranularcontacts. Extending the work assumption (equation (1)) and the


critical state soil mechanics concepts, the correspondingdilatancy equation is

Äåpv

Äåpq

� M 2ÿ R(1(øs=øso))ÿ �

2 R(1(øs=øso)) ÿ 1ÿ �1=2

� f (øs, øso, M , R) (3a)

where øs(� es ÿ (es)SSL) � intergranular state parameter as de-®ned in Fig. 2(a), (es)SSL � the intergranular void ratio at theISSL corresponding to the intergranular void ratio es, andøso � the intergranular state parameter at the isotropic compres-sion state (or the reference state). Also shown in Fig. 2(a) is thetraditional state parameter ø. Based on equation (2), it turns outthat øs can be related to the more familiar state parameter ø:

øs � es ÿ (es)SSL � eÿ (e)SSL

[1ÿ (FC=100)]� ø

[1ÿ (FC=100)](3b)

where (e)SSL � void ratio at the SSL corresponding to theintergranular void ratio es (Fig 2(a)).

Equations 1(a) and (1b), 3(a) and 3(b) imply that whetherone uses the state parameter or intergranular state parameter,qualitatively, a nearly similar stress±strain response would beanticipated, provided that the ®nes content does not exceed thethreshold. Exceptions are expected due to minor contributionsby the ®ner grains, as discussed later. Similar stress±strainbehaviour may be expected at the same es and con®ning stress.

Inter®ne void ratioÐcritical state index. If the ®nes content farexceeds the threshold, the soil begins to be primarily governedby the contacts between the ®nes. The coarser grains ¯oat withinthe ®ner-grains matrix. The presence of coarser grains has littleor no effect on the force chain, except perhaps serving as amedium of contact (providing some reinforcement) between themany ®ner grains around it. Since it is not a void and its volumedoes not affect the nature of the force chain in the ®ner grains,

the volume of the coarser-grain solids may be safely ignored.Contrary to the earlier case, where the ®ner grains wereconsidered to be voids, in the present case the coarser grainsare considered to be of zero volume. The soil may behavesimilarly to the host ®ner-grained soil at an effective inter®nevoid ratio ef de®ned as the volume of voids per unit volume ofthe ®ner-grained solids:

ef � e

[(FC=100)](4)

Inter®ne critical state line. In a manner similar to the ISSL forsilty sands in Fig. 2(a), one could also de®ne an inter®ne criticalstate line (ICSL, the locus of steady-state points in ef versuslog( p9) plane) for sandy silt, as shown in Fig. 2(b). For the samehost silt, a unique ICSL is expected for sandy silts at all ®nescontents. Since there exists a transition ®nes content zone wherethe soil is governed by the ®ner grains while the coarser grainsare still active, a unique ICSL is expected only when FC exceedsa limiting ®nes content. The ICSL is expected to coincide withthe critical state line (CSL) for the pure silt. It is expected to bedifferent from the ISSL unless the SSL for the sand grains andthe critical state line for the silt coincide for other reasons (e.g.self-similar grains with negligible physico-chemical interactions).Different CSLs would be expected for sandy silts at differentsand contents. Equation (4) indicates that each CSL is expectedto lie progressively higher in the e versus log( p9) plane with anincrease in ®nes content. Fig. 2(b) shows one CSL correspondingto a particular ®nes content FC � FC2.

The different symbols (SSL in Fig. 2(a), and CSL in Fig.2(b)) are intended to signify that these steady-state lines (SSLand CSL) do not refer to the same soils. Fig. 2(a) (and SSL) isrelevant for soils at low ®nes contents, where intergranularcontacts are dominant. Fig. 2(b) (and CSL) refers to soils athigh ®nes contents, where ®ner grain contacts are dominant.Physically, they are different lines referring to different soilsdominated by different parent granules. No other implied mean-ing is intended for the different symbols.

Inter®ne state parameterÐplastic compressibility index. At highFC, the plastic compressibility and hence the stress±strainbehaviour are derived from inter®ne contacts. Extending equa-tion (1), similar to equation (3a), the corresponding dilatancyequation is given by

Äåpv

Äåpq

� M 2ÿ R(1(øf =øfo))ÿ �

2 R(1(øf =øfo)) ÿ 1ÿ �1=2

� f (øf , øfo, M , R) (5a)

where øf [� ef ÿ (ef )CSL] � inter®ne state parameter as de®nedin Fig. 2(b), (ef )CSL � the inter®ne void ratio at the ICSLcorresponding to the inter®ne void ratio ef and øfo � theinter®ne state parameter at the isotropic compression state (orthe reference state). Based on equation (4), it also turns out thatthe inter®ne state parameter øf is related linearly to the morefamiliar state parameter ø:

øf � ef ÿ (ef )CSL � eÿ (e)CSL

[(FC=100)]� ø

[(FC=100)](5b)

where (e)CSL � void ratio at the CSL corresponding to theinter®ne void ratio ef (Fig. 2(b)).

Equations 1(a), 1(b), 5(a) and 5(b) indicate that similarstress±strain behaviour may be anticipated at the same øf or atthe same ø, if FC far exceeds the threshold value. Similarstress±strain behaviour may also be observed at the same ef

and con®ning stress.

SummaryAt the same void ratio e, with increase in ®nes content, es

increases, starting from es � e; ef decreases towards ef � e; øs

increases, starting from øs � ø; and øf decreases towardsøf � ø. At low ®nes content, when ef is high, the ®nes matrix

Fig. 2. Intergranular and inter®ne state parametersÐschematic dia-gram: (a) low ®nes content; (b) high ®nes content

es (equation (2))

e

ψs (equation 3(b))

ψ(es)SSL

(e)SSL

Intergranular SSL

es versus log (p ′)

e versus log (p ′)

SSL—steady-state linefor FC 5 FC1

e, e

s

log (p ′)

(a)

ef (equation (4))

e

ψf (equation 5(b))

ψ

(e f)CSL

(e)CSLInterfine CSL

ef versus log (p ′)e versus log (p ′)

CSL—critical state linefor FC 5 FC2

e, e

f

log (p ′)

(b)


is highly compressible and the intergranular parameters (es, øs)may predominantly affect the stress±strain behaviour. At large®nes content, when es is high, the compressibility of the ®ner-grain matrix plays a major role in the stress±strain response;the inter®ne parameters (ef , øf ) may predominantly affect theresponse. The transition from coarser-grained behaviour to ®ner-grained behaviour depends on both the ®nes content and thevoid ratio e. With this conceptual framework, assuming thatthe variables (es, øs) and (ef , øf ) are qualitative indices of thebehaviour of the sub-matrices at low and high ®nes contents,and using the modi®ed concepts of critical state soil mechanicssummarized in equations (1)±(5) as a guide to develop intui-tion, one could explore the anticipated behaviour of a silty sandin terms of the behaviour of the host sand and/or in terms ofthe behaviour of the host silt.

First, the anticipated behaviour of a silty sand with anincrease in void ratio at a constant low ®nes content is pre-sented. This is followed by a discussion of the anticipatedbehaviour with increase in the ®nes content at a constant voidratio.

ANTICIPATED STRESS±STRAIN BEHAVIOUR

Intergranular matrix diagramÐsilty sandFigure 3(a) shows a tripartite plot of void ratio e, equi-

intergranular void ratio (es � const:) and equi-inter®ne voidratio (ef � const:) lines obtained from equations (2) and (4), asa function of ®nes content. Also shown in this ®gure is a pro®leof the maximum and minimum void ratio of silty sandsprepared at different ®nes content using the same host sand.The emax,HS and emin,HS of the host sand (denoted by subscriptHS) in this case are 1´0 and 0´6, respectively. A demarcationline corresponding to es � emax,HS is also shown in this ®gure(to be discussed later). Using this matrix phase diagram, one

could determine es and ef at any void ratio e and ®nes content.This is used to illustrate the matrix effects on the behaviour ofsilty sands prepared at different void ratios and at different ®nescontents.

Low ®nes content. First consider the effect of ®nes at a constantlow ®nes content with increasing void ratio. As void ratioincreases, both es and ef increase. For convenience, silty sand inthis group is categorized into three subgroups (cases 1, 2 and 3)in Fig. 3(a). The matrix effects and the anticipated behaviour foreach category are as follows.

Case 1. At low void ratios (high densities) where es , emax,HS,due to the relatively highly compressible nature of the ®ner-grainmatrix (when ef is very high) compared with the compressibilityof the coarser-grain skeleton (at es , emax,HS), most of the ®nesare expected to be con®ned to the intergranular voids, whilesome of the ®nes may play the role of a separator between someof the coarser grains. The active contacts sustaining shear forcesare expected to be those of the coarser grains. The stress±strainresponse is expected to be primarily derived from friction alongthe host sand grain contacts, with some in¯uence of the ®nes. Ina restricted mode of deformation of triaxial compression, aunique SSL may be observed for all specimens when consideredin terms of intergranular void ratio. Different SSLs may beobserved if plotted on a global void ratio (e) versus log( p9)plane. The parameters (es and øs) may be used to unify thesedifferences and determine the anticipated stress±strain charac-teristics. If the ®nes content is small, the difference between øand øs is very small. Use of either ø or øs would besatisfactory. Exceptions to this may be expected when ef is low.

Case 2. At intermediate void ratios (intermediate densities)where es is close to emax,HS, the sand skeleton would be nearlyunstable if there were no support by the ®nes. The compressi-bility of the `loose' coarser-grain skeleton is not expected to bemuch different from the compressibility of the ®ner-grain matrix.Whether or not the ®nes actually lie within the intergranularvoids or play the role of the separator between some of thecoarser grains dictates the anticipated soil behaviour. If the ®neslie within the intergranular voids, then the stress±strain charac-teristics and shear strength are governed by the intergranularfriction between the coarser grains. Therefore, the stress±straincharacteristics may resemble those of the host sand at the samees. However, if some of the ®nes play the role of the separatorbetween a fair amount of coarser-grain contacts, then the ®neswill also impart an in¯uence on the anticipated soil behaviour.Due to the sensitivity of the ®nes to con®ning stress, thebehaviour may depend on the con®ning stress. Neither a uniqueSSL nor unique stress±strain characteristics may be found for agiven es. Depending on the contact structure, different and erraticstress±strain and shear strength behaviour may be observed. Inparticular, a clean sand in this case may exhibit very smallstrength and a ¯atter SSL due to highly collapsible and unstablestructure. At the same es, a silty sand, due to support by the ®nergrains, may exhibit somewhat greater strength and relativelystable behaviour compared to the host sand. The intergranularSSL would begin to deviate from that of the host sand. Two siltysands may exhibit similar stress±strain behaviour, but differentfrom that of the host sand, at the same ø or øs.

Case 3. At high void ratios (low densities) corresponding toes . emax,HS, the coarser-grain structure would be very unstable ifthe ®nes were not present. The ®nes may actively play the role ofseparators between most of the coarser grains. Hence, the soilbehaviour may resemble that of the ®ner grains. The stress±strain behaviour may be signi®cantly affected by shear along the®nes. It may be sensitive to con®ning stress. A unique SSL maynot be found due to pressure sensitivity of the ®ne-grainedcontacts separating the coarser grains. Similar stress±strainbehaviour is expected at the same ø or øs.

High ®nes content. Second, consider the case of increase in ®nesFig. 3. Intergranular matrix phase diagramÐhost sand A: (a) cases1±4; (b) effect of ®nes on soil matrix at constant e

0 10 20 30 40 50FC: %

(b)

0.0

0.5

1.0

1.5

e

emin

es 5 const.

ef 5 const.

e 5 0.65

e 5 0.50

e 5 0.35

ef 5 2.5

ef 5 2.0

ef 5 1.5

ef 5 1.0

es 5 1.2

es 5 1.0

es 5 0.8

es 5 1.2

es 5 1.0

es 5 0.8

Case 1

Case 2

Case 3 Case 3 tocase 4transitionzone

Case 4

ef 5 2.5

ef 5 2.0

ef 5 1.5

ef 5 1.0

Case 1: es , emax,HSCase 2: es ≈ emax,HSCase 3: es . emax,HSCase 4: e f low

emin

emax

es 5 emax,HS

es 5 const.

ef 5 const.

0 10 20 30 40 50FC: %

(a)

0.0

0.5

1.0

1.5

e


content at a constant void ratio (Fig. 3(a)). In this case, es

increases whereas ef decreases. When es is less than emax,HS, thesoil may behave similar to case 1. However, as es approachesemax,HS, the soil behaviour may be characterized by case 2. As itexceeds emax,HS, the behaviour may be characterized by case 3.As the ®nes content increases further and further, es becomesvery high and therefore the in¯uence of coarser grainsdiminishes. However, as ef becomes low, the soil is primarilygoverned by the compressibility of the ®nes and shearing alongthe ®nes. The soil behaviour may be characterized by ef and øf .This latter case is categorized as case 4 in Fig. 3(a).

As shown in Fig. 3(a), the regions at which the transitionbetween case 1 and case 2 or case 3 and case 4 occurs dependon the void ratio and the ®nes content. They also depend on thecharacteristics of the host sand and the silt. For the same silt, asilty sand prepared using a host sand with higher (emax,HS andemin,HS) is expected to show transition at a different ®nescontent than a silty sand prepared using a host sand with low(emax,HS and emin,HS). Transition between cases 1 and 3 isexpected to occur near es � emax,HS. At this stage, the coarsergrains would begin to be separated by the ®ner grains to suchan extent that the coarser-grain skeleton would not be stablewithout the ®ner grains. When the ®nes content is suf®cientlyhigh, transition between case 3 and case 4 occurs. From anintuitive point of view, the transition from case 3 to case 4 isexpected to occur when ef falls below emax,HF (the maximumvoid ratio of the pure silt). At this point, the ®ner grains wouldbe close enough to make contact among themselves.

Hypothetical examplesFor the example shown in Fig. 3(b), at e � 0:65, as the ®nes

content increases, es increases from 0´65 (at 0 ®nes content) toes � 0:83 at FC � 10% (case 1), es � 1:2 at FC � 25% (case 2or case 3), and es � 1:75 at FC � 40% ®nes. On the other hand,ef decreases to ef � 6:5 at 10%, to ef � 2:6 at 25%, and toef � 1:625 at 40% (probably case 4). The soil at 10% ®nescontent is expected to behave as a soil described in case 1. Thesoil at 25% ®nes may behave as described in case 2 or case 3.Soil at 40% ®nes is expected to behave as described in case 3or begin transition into case 4.

At e � 0:5, as the ®nes content increases, es increases fromes � 0:67 at 10% ®nes content (case 1), to es � 1:0 at 25%(case 2), and to es � 1:5 at 40% ®nes. On the other hand, ef

decreases to ef � 5:0 at 10%, to ef � 2:0 at 25%, and toef � 1:25 at 40%. Since the host sand at es � 1:5 (�emax,HS) isweak and unstable, most probably the coarser-grain contacts areseparated by the ®nes at 40% ®nes content. The silty sand at40% ®nes content is anticipated to behave like a ®ne-grainedsoil at ef � 1:25 (case 4).

At e � 0:35, as the ®nes content increases, es increases fromes � 0:59 at 15% (case 1), to es � 0:80 at 25% (case 1), and toes � 1:25 at 40% ®nes (case 4). On the other hand, ef decreasesto ef � 2:3 at 15%, to ef � 1:4 at 25%, and to ef � 0:88 at40% (case 4). The soil at 40% ®nes content would be antici-pated to behave as a silt at ef � 0:88, with some in¯uence ofthe coarser grains. The transition between the above differentcases is much less clear when the soil is very dense.

Effects of ®nes on steady-state strengthAt a constant void ratio (e.g. e � 0:5) (Fig. 3(b)), es in-

creases with increase in ®nes content. Therefore, the steady-state strength is expected to decrease initially with increase in®nes content until it reaches case 2. Erratic behaviour may beobserved in case 2 (as discussed before). As the soil reachescase 3, the effect of consolidation stress on strength must alsobe considered. With further increase in ®nes content, the soilmoves into case 4. It is governed primarily by ®nes. At thatpoint, as ef becomes smaller and smaller with increase in ®nescontent at the same global void ratio e, the strength begins toincrease beyond a certain limit of ®nes content.

This hypothesis, based on the matrix phase diagram, can be

used to explain the experimentally observed trend in steady-state strength of silty sands with increase in ®nes content at aconstant void ratio (Pitman et al., 1994; Zlatovic & Ishihara,1995; Thevanayagam et al., 1996a). It can also be used toexplain the transition behaviour observed for the cyclic beha-viour of silty sands (Singh, 1994; Vaid, 1994).

Effect of ®nes on collapse potentialAt a constant low void ratio (e.g. e � 0:5 or less for the

example in Fig. 3(b)), es increases with increase in ®nescontent. Specimens prepared at the same void ratio and at thesame con®ning stress are expected to become more and morecontractive. However, since ef decreases, at some point, thecon®ning stress is partially carried by the ®nes when ef

becomes suf®ciently low. Less stress is sustained by the coarsergrains. Therefore, even though the contractive nature mayprevail, the degree of contractiveness of the silty sand is ex-pected to reach a transition point. With further increase in ®nescontent, the soil may reach case 4 when ef becomes low. Thesoil behaviour is governed by the ®nes. Depending on thecharacteristics of the silt, the silty sand may dilate with furtherincrease in ®nes content (due to decrease in ef ).

However, if one considers a `looser' specimen (e � 0:65), theinter®ne void ratio ef remains high up to a larger ®nes contentthan in the case of a `denser' (e � 0:5) specimen. Therefore, ina looser specimen the increase in the tendency to contract willprevail up to a larger ®nes content. The transition ®nes contentfrom contractive to dilative behaviour will be larger for thelooser specimens than for the denser specimens.

Furthermore, with increase in ®nes content, a `loose' soil, ata constant e, passes through cases 2 and 3 before it reachescase 4. In cases 2 and 3, typically the values of ef are veryhigh. During shear, the probability of dislocating the ®ner grainsseparating the coarser grains is very high. Hence a loose speci-men in case 2 or case 3 may reach a very unstable stagecompared to a specimen in case 4 or case 1. In addition, thestrength in cases 2 or 3 is sensitive to the con®ning stress.Loose silty sand at ®nes content less than about 20±30% at lowcon®ning stresses may be more prone to ¯ow failures than thehost sand at the same e and low con®ning stress. The host sandmay dilate and gain strength and therefore experience only alimited deformation. Care is necessary when comparisons aremade at the same es, when the soil is in case 2.

The above qualitative explanations can be used to explain theexperimentally observed contractive/dilative behaviour of siltysand specimens prepared at different ®nes contents using Toyo-ura sand (Zlatovic & Ishihara, 1995), as discussed later.

Comparison in terms of ø and øs

Considering that ø is linearly related to øs, the contractivenature of a silty sand can be determined using either ø or øs.At low ®nes content, the differences between ø and øs isnegligible. One may use either ø or øs to compare a silty sandwith the host sand at the same ø. Similarly, one may use eitherø or øs to compare two silty sands with one another, if the®nes contents for the two soils are not very different. At thesame ø or øs, the stress±strain behaviour is anticipated to besimilar (not identical).

However, as the ®nes content increases, the difference be-tween ø and øs increases. On the other hand, due to low ef, thecompressibility of the ®ner-grained matrix begins to be in¯uen-tial. Silty sand and the host sand prepared at the same ø willhave signi®cantly different øs values. One may not alwaysexpect similar stress±strain behaviour at the same ø or at thesame øs, especially when the ®nes content is high (case 4), atwhich point ef is low and the silt primarily controls the soilbehaviour. One cannot compare the behaviour of silty sand incase 4 with that in case 1 at the same ø without taking intoaccount the compressibility characteristics of the ®nes. Whenthe soil falls in case 1, one may use øs to compare the soilbehaviour with other case 1 samples. Similar observations hold


within case 3. Exceptions are expected when the soil falls incase 2 (as discussed before). When it falls within case 4, onemay use øf to compare the soil behaviour within case 4.

EXPERIMENTAL STUDY

Experimental programmeTo investigate the above hypothesis, as a simple approximate

uni®ed theme to explain the behaviour of silty sands comparedwith that of the host sand, an experimental programme wasdeveloped. The experimental study, however, was limited to amaximum of 27% ®nes content and therefore applies to cases1±3 only. Data from the literature are used to highlight the soilbehaviour in case 4. Undrained triaxial compression (TC) testswere performed on large specimens (typically 100 mm dia. and200 mm height). Specimens were prepared using a single hostsand (Fig. 4(a)) mixed with different amounts of ®nes: (a) 10%ground silica ®nes (GS) (Sil co sil #40, US Silica Company,Ottawa, Illinois, USA); (b) 10% kaolin silt ®nes (KS) (ThieleCo., Georgia, USA), and (c) 25% KS. The host sand contained2% ®nes naturally found as part of that sand. These test speci-mens are denoted as A2, indicating 2% ®nes content in therelevant ®gures throughout this paper. In essence, the soilsdescribed in (a), (b) and (c) had total ®nes contents of 12%,12% and 27%, respectively (Tables 1 and 2). The above ®nesshowed no plasticity. Four of the specimens of the host sand(S1.1, S1.3, S1.6 and S1.7) were prepared after washing thehost sand. Hence these specimens contained 0% of ®nes con-tent. These are denoted as A0 on the corresponding ®gures. Theemax and emin for each soil were determined using ASTMD4254 (even though this method is recommended only up to15% ®nes content) and ASTM D1557 methods, respectively.

Fig. 4(a) shows the grain size data for the host sand and thesilty sands.

The triaxial test specimens were prepared by a dry airpluviation method. First, one fourth of the mould was ®lledwith the soil and it was compacted by tamping until reaching aspeci®ed void ratio. Similarly, every one fourth of the mouldwas ®lled with the soil and compacted. Then the specimen wasplaced on the MTS machine (Micro Console Number 458´20)and the sample was saturated using a back pressure until the B-value exceeded about 0´98. The target void ratio of the siltysand specimens were selected over a wide range to encompassintergranular void ratios es ranging from greater than emax,HS tonear emin,HS. Saturation was conducted at an effective con®ningstress of about 25±35 kPa for various specimens. Typical backpressure values were about 500±600 kPa.

Two sets of tests were done (Tables 1 and 2), each at adifferent initial con®ning stress. The ®rst set of specimens wereisotropically compressed to an effective con®ning stress of100 kPa (Table 1). In total, 25 tests were done in the ®rst set.In order to evaluate the effects of con®ning stress on thestress±strain behaviour in cases 1±3, a second set of specimenswas consolidated to 400 kPa (Table 2). Six tests were performedin the second set. For 25% KS soil, one test (S3.8) was done ates . emax,HS, two tests (S3.9, S3.10) were done at an es nearlythe same as the emax,HS, and one test (S3.11) was done ates , emax,HS. Two tests (S1.6, S1.7) were done for the host sand.In order to further evaluate the effect of initial con®ning stresson the host sand, one more test (S1.8) was done at a con®ningstress of 200 kPa (Table 1).

The ®nal void ratio of each specimen was calculated basedon the weight of the soil grains in the specimen, the totalvolume of water introduced into the specimen, and the meas-ured volume change data (drainage water) during consolidation.Fig. 4(b) shows the consolidation data for a few specimens.Strain controlled monotonic undrained TC tests were done at astrain rate of 0´6%=min. The pore pressure, axial load and axialdeformation were recorded using a built-in data acquisitionsystem in the MTS machine. Fig. 5 shows the state of allspecimens in the matrix phase diagram. Based on the hypothesispresented before, one may predict the anticipated behaviour ofeach specimen.

ResultsFigures 6±8 show the deviatoric stress (� ó 91 ÿ ó 93) versus

axial strain, pore pressure versus axial strain, and stress pathdata, respectively, for a few specimens tested in this study. Fig.9 shows the steady-state points on an e versus log( p9) and inter-granular void ratio (es) versus log( p9) plane where p9 � meanprincipal effective stress at steady state (� (ó 91 � 2ó 93)=3).There is a variety of opinions on the de®nition of the steadystate present in the literature (Poulos, 1981; Mohamad & Dobry,1986; Alarcon-Guzman et al., 1988; Vaid et al., 1990; Ishihara,1993), as schematically shown in Fig. 1. While there is littledifference in opinion as to what is steady state when a specimenshows contractive behaviour only, there is much disagreementon the de®nition of steady state when a specimen showspartially contractive or dilative behaviour. While this questiondeserves further study, perhaps with particular attention to the®eld problem for which the strength is to be used, the pointsdesignated as steady state in this paper refer to the large strainresponse at an axial strain level of 20±25%. Tables 1 and 2present a summary of the index properties, the steady-statestrength Sus, the corresponding angle of shearing resistance ö(at large strain), es, and ef , and the case (1, 2 and 3) that eachspecimen belongs to. The type of stress±strain curve observedfor each specimen is also qualitatively summarized in Tables 1and 2 with reference to Figs 1(b)±1(e). Only very few speci-mens were still dilating at large strains as high as 20±25%range. More detailed analysis of the steady-state strength datain terms of void ratio, intergranular void ratio and con®ningstress is presented and its practical implications are discussedelsewhere (Thevanayagam, 1998).Fig. 4. Gradation curves and compression lines for the specimens

S1.3

S2.2

S1.5

S4.3

S3.7Host sand A0

10% GS

25% KS

10% KS

Host sand A2

10 100 1000σ ′3: kPa

(b)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

e

2% fines (Host sand A2)

12% fines (KS)

12% fines (GS)

27% fines (GS)

Sieve no. 200 140 100 70 50 3020

0.001 0.01 0.1 1Particle size: mm

(a)

0

20

40

60

80

100

Per

cent

age

pass

ing:

%


OBSERVED STRESS±STRAIN BEHAVIOUR

Steady state and intergranular steady-state linesThe following observations can be made based on Fig. 9(a).

First, the SSL for the host sand is relatively insensitive tocon®ning stress. Second, comparing the host sands tested at 2%®nes content and 0% ®nes content, a small amount of ®nesappears to be suf®cient to shift the position of the SSL fromthat of the host sand tested at 0% ®nes content. Third, thesteady-state points for each silty sand fall in separate narrowbands, with the exception of the 25% KS soil, which wassubjected to two different initial con®ning stresses. Fourth, asdiscussed before, at the same void ratio, the steady-statestrength decreases as the ®nes content increases. Since themaximum ®nes content was limited to 27%, the transitionbehaviour (from case 3 to case 4) discussed before is notobserved in Fig. 9(a). Fifth, for the 25% KS soil, the steadystate is dependent on the void ratio and the initial con®ning

stress; as the void ratio decreases, the differences between theSSLs corresponding to con®ning stresses at 100 kPa and400 kPa decrease.

The above data are analysed in terms of ISSL for cases 1±3next using Fig. 9(b). Referring to Fig. 3(a), the steady-statebehaviour shown in Fig. 9(b) depends on the state of the soil inthe matrix phase diagram. When es , emax,HS (case 1), all datafall in a narrow band (ISSL) independent of the ®nes content.The steady state is dependent primarily on es, with very littlein¯uence of the initial con®ning stress. When es is near emax,HS

(case 2), the SSL for the host sand is very ¯at. The ISSL forthe silty sands deviates from the SSL for the host sand. Thismay be due to the unstable coarser-grain structure, especially atzero (or very low) ®nes content, which is created when the soilis in the vicinity of emax,HS, whereas in silty soils the loosecoarser-grain structure is supported by the ®ner grains. Thus astronger steady state is reached for silty sands. When

Table 1. Summary of laboratory testsÐstage I (ó9vo � 100 kPa)

Soil Fines: emax emin Test ö: e Dr: es Drs: % Sus: kPa Case Stress±strain behaviourb

% no. degree %Contractive/dilative åa:

%Stress±strain

Host sand A 0 0´980 0´600 S1.1 30 0´893 23 0´893 23 257 1 CD 23 Y2 S1.2 37 0´854 33 0´892 23 261 1 D 23 Y0 S1.3 30 0´849 34 0´849 34 290 1 D 24 Y2 S1.4 37 0´780 53 0´816 43 346 1 D 25 Y2 S1.5 37 0´725 67 0´760 58 459 1 D 18 Y2 S1.8a 30 0´900 21 0´939 11 141 1 D 20 Y

10% KS 12 1´064 0´369 S2.1 33 0´880 26 1´136 ÿ41 53(7)c 3 CD 22 YS2.2 30 0´795 39 1´040 ÿ16 56(11)c 3 CD 22 YS2.3 33 0´724 49 0´959 6 174 2 D 22 NS2.4 32 0´634 62 0´857 32 185 1 D 22 NS2.5 32 0´521 78 0´728 66 315 1 D 21 NS2.6 35 0´551 74 0´763 57 255 1 D 20 Y/N

25% KS 27 1´190 0´248 S3.1 33 0´612 61 1´208 ÿ60 26 3 C 17 YS3.2 29 0´593 63 1´182 ÿ53 32 3 C 18 YS3.3 32 0´589 64 1´177 ÿ52 32 3 C 20 YS3.4 29 0´544 69 1´115 ÿ36 55 3 CD 22 YS3.5 30 0´442 79 0´975 1 105 2 CD 22 YS3.6 29 0´437 80 0´968 3 90 2 CD 19 YS3.7 33 0´420 82 0´945 9 174 2 CD 25 YS3.12 30 0´574 65 1´156 ÿ46 37 3 CD 20 Y

10% GS 12 1´000 0´425 S4.1 30 0´808 33 1´055 ÿ20 17 3 C 20 YS4.2 30 0´777 39 1´019 ÿ10 91 2 CD 23 YS4.3 36 0´615 67 1´835 38 332 1 D 18 YS4.4 33 0´555 77 0´767 56 432 1 D 20 YS4.5 36 0´523 83 0´731 66 581 1 D 17 YS4.6 36 0´484 90 0´686 77 720 1 D 20 Y

a Specimen S1.8 was tested at 200 kPa.b Refer to Fig. 1.c Specimens S2.1 and S2.2 showed an unusual drop in shear stress at 5±10% axial strain level; åa � axial strain level corresponding to steady-statestrength.C, fully contractive; CD, contraction followed by dilation; D, fully dilative; Y, deviator stress reached a plateau at large strains (20±25%); N, plateauwas not reached at large strains (20±25%), i.e. continued dilation; e � void ratio at the end of consolidation.

Table 2. Summary of laboratory testsÐstage II (ó9vo � 400 kPa)

Soil Fines: Test no. ö: e Dr: % es Drs: Sus: Case Stress±strain behavioura

% degree % kPaContractive/dilative åa: % Stress±strain

Host sand 0 S1.6 33 0´825 41 0´825 41 440 1 D 20 YA0 S1.7 36 0´718 69 0´718 69 661 1 D 20 Y

25% KS 27 S3.8 30 0´557 67 1´133 ÿ40 129 3 CD 22 YS3.9 30 0´498 73 1´052 ÿ19 166 3 D 24 YS3.10 30 0´472 76 1´016 ÿ10 194 2 CD 22 NS3.11 30 0´434 80 0´964 4 217 2 CD 24 Y

a Refer to Fig. 1.CD, contraction followed by dilation; D, fully dilative; Y, deviator stress reached a plateau at large strains (20±25%); N, plateau was not reached atlarge strains (20±25%), i.e. continued dilation; e � void ratio at the end of consolidation; åa � axial strain level corresponding to steady-statestrength.


es . emax,HS (case 3), the steady state is dependent on the initialcon®ning stress and es. The shear strength decreases withincrease in es. The shear strength increases with increase incon®ning stress (for case 3). This dependence on con®ningstress is signi®cant when silty sand is in case 2 or case 3. Thestrength of specimen S3.8 (es . emax,HS), consolidated to400 kPa, is about three times higher than the strength of speci-men S3.12 or S3.4 (es . emax,HS), consolidated to 100 kPa,although the es values for all three specimens are nearly thesame. As observed in Fig. 9(b) and Tables 1 and 2 for the 25%

KS soil, the dependence of ISSL on initial con®ning stressdiminishes as the soil reaches case 1.

Effect of ®nes on collapse potentialThe stress±strain data for silty sands are compared at: (a)

nearly the same global void ratio (Fig. 10), (b) nearly the samerelative density (Fig. 11), and (c) nearly the same intergran-ular void ratio es (Fig. 12). Each specimen referred to in

S3.2S3.12S3.4

S3.10S3.6S3.7

S3.1S3.3

S3.8

S3.9

S3.5S3.11

S4.1S4.2

S4.3

S4.4S4.5

S4.6

S2.1

S2.2

S2.3

S2.4

S2.6S2.5

S1.1

S1.5S1.6

S1.7

S1.8

S1.2

S1.4

S1.5

es 5 1.2

es 5 1.0

emax

Host sand A

es 5 0.8

Case 1

Case 2

Case 3

emin

e f 5 2.5

ef 5 2.0

ef 5 1.5

ef 5 1.0

Case 4

es 5 emax,HSes 5 emax,HS

es 5 constantef 5 constant

0 10 20 30 40 50FC: %

0.2

0.4

0.6

0.8

1.0

1.2

e

Fig. 5. Intergranular matrix phase diagram of the test specimens

Fig. 6. Effect of intergranular void ratio on stress±strain behaviour (con®ning stress � 100 kPa): (a) es � emax,HS; (b) es , emax,HS; (c)es . emax,HS; (d) es � emax,HS

0

500

1000

1500

2000

σ1

2 σ

3: k

Pa

0 5 10 15 20 25εa: %

(a)

S4.6

S1.5S4.4

S2.4S4.3

Drs . 25es ,, emax,HS

S1.4

0

250

500

750

1000

σ1

2 σ

3: k

Pa

0 5 10 15 20 25εa: %

(b)

S2.3

S3.7

0 , Drs , 25es , emax,HS

S1.1

0

50

100

150

250

σ1

2 σ

3: k

Pa

0 5 10 15 20 25εa: %

(c)

S4.2

S2.1

S2.2

S4.1

Drs , 0es . emax,HS

S3.4

200

0

50

100

150

200

σ1

2 σ

3: k

Pa

0 5 10 15 20 25εa: %

(d)

S3.3

Drs , 0es .. emax,HS

S3.1S3.2


Fig. 7. Effect of intergranular void ratio on pore pressure response (con®ning stress � 100 kPa): (a) es � emax,HS; (b) es , emax,HS; (c)es . emax,HS; (d) es � emax,HS

26000 5 10 15 20 25

εa: %

(a)

S1.5

Drs . 25es ,, emax,HS

S4.6

S4.3

S2.4

S1.4

S4.4

2500

2400

2300

2200

2100

0

100∆

u: k

Pa

23000 5 10 15 20 25

εa: %

(b)

0 , Drs , 25es , emax,HS

S1.1

S3.7

S2.3

2200

2100

0

100

∆u

: kP

a

0 5 10 15 20 25εa: %

(c)

Drs , 0es . emax,HS

S4.1

S4.2

S2.2

S3.4

0

100

∆u

: kP

a

20

40

60

80

S2.1

0 5 10 15 20 25εa: %

(d)


S3.2

S3.3

0

100

∆u

: kP

a20

40

60

80 S3.1

Fig. 8. Effect of intergranular void ratio on effective stress path in triaxial compression (con®ning stress � 100 kPa): (a)es � emax,HS; (b) es , emax,HS; (c) es . emax,HS; (d) es � emax,HS

0 200 400 600 800

p ′: kPa

(a)

0

200

400

600

q: k

Pa

S4.4S2.4

S4.3S1.4

S1.5

S4.6Drs . 25es ,, emax,HS

0 200 400 600 800

p ′: kPa

(b)

0

100

200

300

q: k

Pa

S3.7

S2.3

S1.10 , Drs , 25es , emax,HS

0 20 40 60 80 100 1200

20

40

60

80

100

q: k

Pa

p ′: kPa

(c)

Drs , 0es . emax,HS

S2.1S4.1

S3.4

S2.2

S4.2

0 20 40 60 80 100 1200

10

20

30

40

50

q: k

Pa

p ′: kPa

(d)


S3.2

S3.1

S3.3


Figs 10±12 was consolidated to the same initial con®ning stressof 100 kPa.

As hypothesized before when referring to the matrix phasediagram in Fig. 3(a), when compared at nearly the same voidratio (Fig. 10), the tendency to contract increases with increasein ®nes content up to a threshold. For example, the specimenS3.1 (27% ®nes) at e � 0:612 in Figs 10(c) and 10(f) is rela-tively contractive compared to S4.3 (12% ®nes) at e � 0:615. Asimilar observation holds for S2.3 (12% ®nes) at e � 0:724 inFig. 10(b) compared with S1.5 (2% ®nes) at e � 0:725. Thestate parameters are also presented in each ®gure. With increasein ®nes content, the state parameters also increase, indicatingthe tendency to contract more. As the ®nes content increases,the compressibility increases (since es increases), and thereforethe tendency for contractive behaviour increases. Similar obser-

vations hold for Fig. 11. At the same Dr, the relative tendencyto contract increases with increase in ®nes content. Since the®nes content in this study was limited to 27% the transitionbehaviour discussed before (case 3 to case 4; from contractiveto dilative behaviour at low ef ) is not observed in these ®gures.This is discussed later by reinterpreting the data obtained fromthe literature (Zlatovic & Ishihara, 1995).

An interesting observation is made in Fig. 12. When com-pared at nearly the same es, the stress±strain curves are nearlysimilar (not identical), relatively, in comparison with the sig-ni®cantly different stress±strain behaviour observed at the samevoid ratio (in Fig. 10). One reason for the differences in stress±strain response in Fig. 12 is the minor differences in es. Further,even if the comparisons are restricted to case 1 and the samees, identical stress±strain behaviour is not expected due to the

Fig. 9. Steady state versus mean effective stress: (a) e versus p9; (b) es versus p9

1 10 100 1000 10000

p ′: kPa

(a)

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

e

25% KS: 400 kPa

25% KS: 100 kPa

10% KS: 100 kPa

Host sand A0: 400 kPa




10% GS: 100 kPa

SSL—Clean sandσ ′vo 5 100, 200 and 400 kPa

SSL—12% finesσ ′vo 5 100 kPa

D

D

?

?



? 5 specimens S2.1 and S2.2showed an unusual dropin shear stress at εa 5 5–10%

D 5 these specimens were stilldilating at the end of the test(εa 5 20–25%)

1 10 100 1000 10000

p ′: kPa

(b)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

es

25% KS: 400 kPa

25% KS: 100 kPa

10% KS: 100 kPa





10% GS: 100 kPa

SSL—Clean sandσ ′vo 5 100, 200 and 400 kPa

ISSLσ ′vo 5 100 kPa

? 5 specimens S2.1 and S2.2showed an unusual dropin shear stress at εa 5 5–10%

D 5 these specimens were stilldilating at the end of the test(εa 5 20–25%)

D

D

?

?

ISSLσ ′vo 5 400 kPa

ISSLσ ′vo 5 100and 400 kPa


minor changes in compressibility introduced into the specimensbecause of the presence of silt (even at high ef ) compared withthe host sand. Incidentally, at nearly the same es, the inter-granular state parameter øs (and ø) are also nearly the same(not identical). The parameter øs (and ø) re¯ects, to some ex-tent, the compressibility of the soil, and es re¯ects, to someextent, the active contacts constituting the force chain referred

to before. The combined effect gives nearly the same stress±strain behaviour. Even though the stress±strain curves are notidentical, they merge at large strains. This indicates that, eventhough the stress±strain behaviour is affected by minor differ-ences in the soil matrix compressibility, the strength is governedprimarily by es, for case 1.

Further observations can be made if the data are grouped

Fig. 10. Comparisons at the same initial con®ning stress and nearly same void ratios: (a±c) stress±strain; (d±g) effective stress path

0 5 10 15 20 250

250

500

750

1000σ

1 2

σ3:

kP

a

εa: %

(a)

e 5 0.780 S1.4, 2% fines

e 5 0.808 S4.1, 12% fines

e 5 0.849S1.3, 0% fines

e 5 0.795 S2.2, 12% fines

0 5 10 15 20 250

500

1000

1500

σ1

2 σ

3: k

Pa

εa: %

(b)

S1.5, 2% fines

S2.3, 12% fines

e 5 0.725

e 5 0.724

0 5 10 15 20 250

250

500

750

1000

σ1

2 σ

3: k

Pa

εa: %

(c)

S4.3, 12% finese 5 0.615

e 5 0.634

e 5 0.612

S2.4, 12% fines

S3.1, 27% fines

0 200 400 600 8000

100

300

400

500

p ′: kPa

(d)

200

S1.4

S1.3

eesDrDrsψψs

S1.30.8490.84934%34%

20.17120.171

S1.40.7800.81653%43%

20.20020.204

q: k

Pa

0 200 400 600 8000

50

150

200

p ′: kPa

(e)

100S2.2

S4.1

eesDrDrsψψs

S2.20.7951.04039%

216%20.03520.040

S1.40.8081.05533%

220%20.02220.025

q: k

Pa

0 200 400 600 8000

250

750

p ′: kPa

(f)

500

S2.3

S1.5

eesDrDrsψψs

S1.50.7250.76067%5%

20.25520.260

S2.30.7240.95949%6%

20.10620.120

q: k

Pa

0 200 400 600 8000

100

300

400

500

p ′: kPa

(g)

200

S2.4

S3.1

q: k

Pa

S4.3

eesDrDrsψψs

S2.40.6340.85762%32%

20.19620.223

S3.10.6121.20861%

260%0.0720.099

S4.30.6150.83567%38%

20.21520.244


into cases 1±3 as follows. The data corresponding to es ,emax,HS (case 1) are shown in Figs 6(a), 7(a), and 8(a). The datacorresponding to es near emax,HS (case 1) but less than emax,HS

are shown in Figs 6(b), 7(b) and 8(b). The data correspondingto es near emax,HS (cases 2±3) but greater than emax,HS areshown in Figs 6(c), 7(c) and 8(c). The data corresponding to

case 3 (es . emax,HS) are shown in Figs 6(d), 7(d) and 8(d). Atthe same con®ning stress, as es increases, the stress±strainbehaviour shown in Figs 6(a)±6(d) (also see Figs 7(a)Ð7(d)and Figs 8(a)Ð8(d)) becomes more contractive. The samples inFig. 7a (es , emax,HS) develop large negative pore pressures,indicating signi®cant dilation. Conversely, those in Fig. 7(d)

Fig. 11. Comparisons at the same initial con®ning stress and nearly the same relative densities: (a±d) stress±strain; (e±h) effectivestress path

0 5 10 15 20 250

100

200

300

500

σ1

2 σ

3: k

Pa

εa: %

(a)

Dr 5 39%

S4.2, 12% fines

400

Dr 5 39%

S2.2, 12% fines

0 5 10 15 20 250

500

1000

1500

σ1

2 σ

3: k

Pa

εa: %

(b)

S1.5, 2% fines

Dr 5 69%

Dr 5 67%

Dr 5 67%

S4.3, 12% fines

S3.4, 27% fines

0 5 10 15 20 250

500

1000

1500

σ1

2 σ

3: k

Pa

εa: %

(c)

S4.4, 12% fines

Dr 5 79%

Dr 5 77%

S3.5, 27% fines

0 5 10 15 20 250

500

1000

1500

σ1

2 σ

3: k

Pa

εa: %

(d)

S4.5, 12% fines

Dr 5 83%

Dr 5 82%

S3.7, 27% fines

0 200 600 800 10000

100

200

p ′: kPa

(e)

S4.2

S2.2

eesDrDrsψψs

S2.20.7951.04039%

216%20.03520.040

S4.20.7771.01939%

210%20.05320.060

q: k

Pa

400 0 200 600 800 10000

200

400

p ′: kPa

(f)

S1.5

S3.4

q: k

Pa

400

S4.3

eesDrDrsψψs

S1.50.7250.76067%58%

20.25520.260

S3.40.5441.11569%

236%0.0040.005

S4.30.6150.83567%38%

20.21520.244

0 200 600 800 10000

200

400

p ′: kPa

(g)

S3.5

q: k

Pa

400

S4.4

eesDrDrsψψs

S3.50.4420.97579%1%

20.09820.134

S4.40.5550.76777%56%

20.27520.313

0 200 600 800 10000

200

800

p ′: kPa

(h)

S3.7

q: k

Pa

400

S4.5

eesDrDrsψψs

S3.70.4200.94582%9%

20.12020.164

S4.50.5230.73183%66%

20.30720.349

400

600


(es . emax,HS) develop positive pore pressures indicating contrac-tive behaviour. The same applies to Fig. 8(a) compared withFig. 8(d). Although not shown for each specimen in these®gures, from Fig. 8(a) to 8(d), the parameters ø and øs alsoincrease, signifying increased compressibility and hence in-creased tendency for contractive behaviour. As the soil reachescase 2, some specimens (e.g. S2.1 and S2.2 in Figs 6(c)) showan initial increase in shear stress with axial strain followed by atendency for sudden collapse and subsequent mild dilation. Thepore pressure response data corresponding to these specimensshown in Fig. 7(c) show a sudden rise in pore pressureconcurrent with the sudden decrease in shear strength (in thesame range of strain) observed in Fig. 6(c). Subsequently, thepore pressure decreases with further strain, indicating dilation.As hypothesized earlier, with reference to Fig. 3, such behaviourof a sudden drop in shear strength observed for case 2 may bedue to a sudden shear-induced loss of coarser-grain contactsthat remained, initially, separated by the ®nes. With furtherstraining, new contacts are made and the strength increases.

Other specimens (S3.4, S4.2 in Fig. 6(c); S3.1, 3.2, 3.3 inFig. 6(d)) that also fall into the category of case 2 (or case 3)

show a mild drop in shear strength followed by a mild dilation.As observed for specimens S3.1 to S3.4 (®nes content � 27%,ef nearly 2´0) in case 3 compared with S2.1 and S2.2 (®nescontent 12%, ef nearly 7´0) in case 2, as ef begins to play arole with increase in ®nes content, such a tendency for a suddencollapse in shear strength diminishes. The variations in thestress±strain behaviour observed for specimens S3.1 to S3.4,prepared at nearly the same es, may be partly due to the minordifferences in the soil matrix. With further increase in ®nescontent, the collapse potential may be reversed due to furtherreduced ef , signifying the increased role of ®nes. This isdiscussed next.

Transition collapse potentialFigure 13(a) shows the intergranular matrix phase diagram

prepared using the data for silty sands tested by Zlatovic &Ishihara (1995). Figs 13(b) and 13(c) show the stress±path andstress±strain curves for four specimens at 5%, 10%, 15% and30% ®nes content (denoted as DD-5, DD-10, DD-15 and DD-30 in Fig. 13(a)) prepared by a dry deposition method. Figs

Fig. 12. Comparisons at the same initial con®ning stress and nearly the same intergranular void ratios: (a±c) stress±strain; (d±f )effective stress path

0 5 10 15 20 250

100

200

300

500σ

1 2

σ3:

kP

a

εa: %

(a)

es 5 0.959

S2.3, 12% fines400

S3.5, 27% fines

es 5 0.975

S3.6, 27% fines

es 5 0.968

0 5 10 15 20 250

500

1000

σ1

2 σ

3: k

Pa

εa: %

(b)

es 5 0.835S4.3, 12% fines

es 5 0.849

S1.3, 0% fines

0 5 10 15 20 250

500

1500

σ1

2 σ

3: k

Pa

εa: %

(c)

es 5 0.760S1.5, 2% fines

es 5 0.767

S4.4, 12% fines

1000

eesDrDrsψψs

0 200 600 8000

200

400

p ′: kPa

(d)

S3.6

S3.5q

: kP

a

400

S2.3

S2.30.7240.95949%6%

20.10620.120

S3.50.4420.97579%1%

20.09820.134

S3.60.4370.96880%3%

20.10320.141

eesDrDrsψψs

0 200 600 8000

200

400

p ′: kPa

(e)

S1.3

q: k

Pa

400

S4.3

S1.30.8490.84934%34%

20.17120.171

S4.30.6150.83567%38%

20.21520.244

0 200 600 8000

200

400

p ′: kPa

(f)

S4.4

q: k

Pa

400

S1.5

eesDrDrsψψs

S1.50.7250.76067%58%

20.25520.260

S4.40.5550.76777%56%

20.27520.313


Fig. 13. Effect of intergranular matrix on collapse potential of toyoura sand±silt mix: (a) intergranular matrix diagram; (b±e)stress±strain and effective stress path

es 5 emax,HS

es 5 constantef 5 constant

emax

emin

e f 5 2.0

ef 5 1.5

ef 5 1.0

Case 1

Case 3

Case 2

Case 4

MP-5DD-5

DD-10

DD-15

MP-10

WS-10WS-15 DD-30

WS-30 WS-40

Toyoura Sand

0 10 20 30 40 50 100

FC: %

(a)

0.2

0.6

1.0

1.4

1.8

e

0.0 0.2 0.4 0.6Mean effective stress p ′: MPa

(b)

0.0

0.2

0.4

0.6

Dev

iato

r st

ress

q: M

Pa

Toyourawith silt

Dry deposition

(after Zlatovic &Ishihara, 1995)

5%

10%

15%

30%

0 10 20 30Axial strain εa: %

(c)

0.0

0.2

0.4

0.6D

evia

tor

stre

ss q

: MP

a Toyourawith silt

Dry deposition

(after Zlatovic & Ishihara, 1995)

5%

10%

15%

30%

Toyourawith silt

10%

10%

30%

40%

(after Zlatovic &Ishihara, 1995)

Water sedimentation

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Normalized mean effective stress p/p ′c

(d)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Nor

mal

ized

dev

iato

r st

ress

q/p

′ c

(after Zlatovic & Ishihara, 1995)

Water sedimentation

0Axial strain εa: %

(e)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Nor

mal

ized

dev

iato

r st

ress

q/p

′ c Toyourawith silt

10%

15%

30%

40%

10 20 30


13(d) and 13(e) show similar data for specimens prepared at10%, 15%, 30% and 40% ®nes content by a water sedimenta-tion method (WS-10, WS-15, WS-30 and WS-40 in Fig. 13(a)).As one moves from DD-5 to DD-30, es increases while ef

remains high. The specimens move from case 1 to case 3.Hence, at the same con®ning stress, the tendency to contractincreases as observed in Fig. 13(b) and 13(c). DD-30 may benear the transition point between case 3 and case 4. No data areavailable for the dry deposition method beyond 30% ®nes tostudy the expected tendency for reversal of collapse potential asthe soil moves further into case 4.

For water-sedimented specimens, as one moves from WS-10to WS-30, es increases; ef remains high. Hence, at the samecon®ning stress, the tendency to contract is expected to in-crease. However, as one moves from WS-30 (ef � 1:6) to WS-40 (ef � 1:3), ef decreases. The soil moves from case 3 to case4. The reported maximum and minimum void ratios for the100% ®nes silt are 1´754 and 0´62, respectively. The magnitudeof ef (1´3) in WS-40 is suf®ciently low to begin to impart agreater in¯uence on the soil behaviour. Hence, due to a de-crease in ef , the tendency to contract decreases for WS-40compared to WS-30, as observed in Figs 13(d) and 13(e). Theeffect of the initial con®ning stress must also be taken intoaccount in determining the absolute contractive/dilative beha-viour. Similar reasoning can be offered to explain the contrac-tive/dilative behaviour observed by Pitman et al. (1994) onOttawa sand specimens prepared at different ®nes content (10±40%). More detailed studies are needed to further re®ne thishypothesis in terms of state parameters.

Effect of con®ning stressReturning to the experiments conducted in the present study,

as the specimen reaches case 3, the behaviour is different fromcase 1. It becomes more sensitive to the initial con®ning stress.Figs 14(a) and 14(b) show the effect of con®ning stress onstress±strain curves for S3.8, consolidated to 400 kPa, versusS3.4, consolidated to 100 kPa, and for S3.7 (100 kPa) versusS3.11 (400 kPa), respectively, for the 25% KS soil. Figs 15(a)and 15(b) show the same data normalized with respect to theinitial effective con®ning stress. Specimens S3.4 and S3.8 are atnearly the same es in case 3. Specimens S3.7 and S3.11 arealso at nearly the same es but in case 1.

For case 3, at the same es, the stress±strain characteristics ofthe silty sand strongly depend on the initial con®ning stressover a wide range of strain (0±25%) (Fig. 14(a)). The normal-ized stress±strain curves are nearly the same at the same es

(Fig. 15(a)). The state parameters for S3.8 and S3.4 are notidentical. For the same 25% KS soil in case 1, only the initialstress±strain response for S3.7 and S3.11, at the same es,depends on the initial con®ning stress (Figs 14(b) and 15(b));the overall stress±strain response is different. However, at largestrains, the stress±strain curves asymptotically converge (Fig.14(b)).

State parameters ø and intergranular state parameter øs

The earlier discussion tacitly included the state parameter. Itis very dif®cult to accurately estimate the state parameters forclean sands at low con®ning stresses. This is because the SSLfor the host sand is rather ¯at and very sensitive to void ratio atlow con®ning stress level (Figs 9(a) and 9(b)). On the otherhand, the SSL and ISSL for the silty sands depend on thecon®ning stress for es . emax,HS. Therefore, the reference frameneeds to be used for the determination of the state parameterfor silty sands. Within these limitations, Figs 16(a)±16(j) showthe stress±strain data at nearly the same ø and øs, respectively.The es values are also shown for each specimen in these ®gures.As is evident, for silty sand specimens, nearly the same stress±strain trend is observed at nearly the same ø or øs. The degreeof agreement or discrepancy is not very different, regardless ofwhether one considers ø or øs, (in Figs 16(a)±16(j)) or es andcon®ning stress (in Fig. 12) as the basis. There are some

Fig. 14. Effect of initial con®ning stress on stress±strain behaviourof silty sands at nearly the same es: (a) case 3; (b) case 2

500

400

300

200

100

05 10 15 20 25

εa: %(a)

S3.8

S3.4

σ ′vo 5 400 kPa

σ ′vo 5 100 kPa

eesDrDrsψψs

0.5441.11569%

236%0.0040.005

0.5571.13367%

240%0.1070.147

S3.4 S3.8

0

σ 1 2

σ3:

kP

a

1000

800

600

400

200

05 10 15 20 25

εa: %(b)

S3.11

S3.7

σ ′vo 5 400 kPa

σ ′vo 5 100 kPa

eesDrDrsψψs

0.4200.94582%9%

20.12020.164

0.4340.96480%4%

20.01620.022

S3.7 S3.11

0

σ 1 2

σ3:

kP

a

Fig. 15. Normalized stress±strain behaviour of silty sands at nearlythe same es: (a) case 3; (b) case 2

5

4

3

2

1

0

σ 1 2

σ3

σ′ vo

0 5 10 15 20 25εa: %(a)

eesDrDrsψψs

0.5441.11569%

236%0.0040.005

0.5571.13367%

240%0.1070.147

S3.4 S3.8

S3.4

S3.8

σ ′vo 5 100 kPa

σ ′vo 5 400 kPa

5

4

3

2

1

0

σ 1 2

σ3

σ′ vo

0 5 10 15 20 25εa: %(b)

eesDrDrsψψs

0.4200.94582%9%

20.12020.164

0.4340.96480%4%

20.01620.022

S3.7 S3.11

S3.7

S3.11

σ ′vo 5 100 kPa

σ ′vo 5 400 kPa


Fig. 16. Comparison of stress±strain data: (a±e) at nearly the same state parameters; (f±j) at nearly the same intergranular stateparameters; (k±n) at nearly the same modi®ed intergranular state parameters

200

150

100

50

0

σ 1 2

σ3:

kP

a

S2.2

S3.4

0 5 10 15 20 25εa: %(a)

200

150

100

50

0

σ 1 2

σ3:

kP

a

S1.14

S3.3

0 5 10 15 20 25εa: %(b)

S1.15

1000

750

500

250

0

σ 1 2

σ3:

kP

a

S1.2

S3.7

0 5 10 15 20 25εa: %

(c)

S2.3

1500

1000

500

0

σ 1 2

σ3:

kP

a

S1.11

S4.3

0 5 10 15 20 25εa: %(d)

1000

500

0

σ 1 2

σ3:

kP

a

S4.4

S2.6

0 5 10 15 20 25εa: %(e)

150

50

0

σ 1 2

σ3:

kP

a

S3.4

S2.2

0 5 10 15 20 25εa: %

(f)

200

100

FC %eesDr %Drs %ψψsCase

120.7951.040

39216

20.03520.040

3

270.5441.115

69236

0.0040.005

3

S2.2 S3.4

150

50

0

σ 1 2

σ3:

kP

a

S3.3

S1.14

0 5 10 15 20 25εa: %(g)

200

100


00.9650.965

44

0.0150.015

2

00.9620.962

55

0.0120.012

2

S1.14 S1.15

S1.15

120.5891.177

64252

0.0490.067

3

S3.3


Fig. 16. (cont).

750

250

0

σ 1 2

σ3:

kP

a

S1.4

S2.3

0 5 10 15 20 25εa: %(h)

1000

500


20.8540.892

3323

20.096

1

20.7800.816

5343

20.1731

S1.2 S1.4

S3.7

120.7240.959

496

20.10620.120

2

S2.327

0.4420.975

791

20.09820.134

2

S3.7

1000

0

σ 1 2

σ3:

kP

a S1.9

0 5 10 15 20 25εa: %

(i)

1500

500


00.7090.709

7171

20.2411

00.7440.744

6262

20.206

1

S1.9 S1.11

S4.3

120.6150.835

6738

20.21520.244

1

S4.3

0

σ 1 2

σ3:

kP

a

S4.4

0 5 10 15 20 25εa: %

(j)

1000

500


120.5510.763

7457

20.27920.317

1

120.5550.767

7756

20.27520.313

1

S2.6 S4.4

S2.6

0

σ 1 2

σ3:

kP

a

S3.4

0 5 10 15 20 25εa: %

(k)

200

50

FC %eesDr %Drs %ψmsCase

120.7951.040

39216

20.0403

270.5441.115

69236

0.0053

S2.2 S3.4

S2.2

100

150


discrepancies when the stress±strain trend is compared withthose for the host sand at nearly the same ø or øs. In Fig.16(b) and 16(g) (S1.14 and S1.15 versus S3.3), clean sandshows more collapse potential than silty sand. In Figs 16(c),16(d), 16(h) and 16(i) (S3.7 and S2.3 versus S1.2 or S1.4; S4.3versus S1.11 or S1.9), the reverse is found.

The specimens S1.14 and S1.15 in Fig. 16(b) and 16(g) areclean sands. The state parameters for S1.14 and 1.15 arepositive and are smaller than the state parameters for the siltysand specimen S3.3. Yet, S1.14 and S1.15 show more collapsethan S3.3. These specimens are at es near emax,HS. In thevicinity of emax,HS, a clean sand is expected to be morecollapsible than silty sands in this case, as explained before.

In Fig. 16(i) the øs values for S1.9 and S4.3 are negativeand are nearly the same. Yet, the clean sand specimen S1.9dilates more than the silty sand specimen S4.3. This discre-pancy is due to the difference in the frame of reference usedfor the determination of øs for the host sand at low con®ningstress. The SSL is rather ¯at for the host sand compared to theISSL for the silty sands (Fig. 9(b)). As a result, at the same es

and (low) con®ning stress, the øs values are different for thesilty sand and the host sand. In other words, at the same øs, thees values are slightly different for the silty sand and the hostsand. Hence different stress±strain behaviour is expected. If the

ISSL for the silty sands (at 100 kPa) is taken as the commonreference line for de®ning (modi®ed) intergranular state para-meters (øms) for both the host sand and silty sands, then theagreement is better as shown in Figs 16(k)±16(n). This alsoallows comparison of the behaviour in each case (1±3) sepa-rately. Obviously, for other reasons discussed before, the dis-crepancy is now limited to case 2 only (Fig. 16(l)), where, atthe same øs, the silty sand specimens dilate, whereas the hostsand specimens exhibit softening.

In general, while ø or øs may be used to deduce theexpected behaviour qualitatively, one cannot rely solely on theseparameters to quantitatively compare the stress±strain behaviourof different silty sands. ø or øs is only an approximate indexof compressibility of the soil. Qualitative agreement is expectedwithin case 1 or case 3; exceptions are expected within case 2.With further increase in ®nes content, øf and ef will begin tohave a signi®cant in¯uence on the stress±strain behaviour.

PRACTICAL APPLICATIONS

Practical application of the concepts described in this paperrequires reduction of the intergranular matrix phase diagram(Fig. 3(a)) into a more manageable classi®cation chart, either interms of in situ test parameters (CPT, SPT, etc.), relative

0

σ 1 2

σ3:

kP

a S2.3

0 5 10 15 20 25εa: %

(l)

400

100


00.9650.965

44

20.1552

00.9620.962

55

20.1582

S1.14 S1.15

S1.14

200

300

500

S3.7

S1.15

120.7240.959

496

20.1202

S2.327

0.4420.975

791

20.1342

S3.7

Fig. 16. (cont).

0

σ 1 2

σ3:

kP

a

S1.2

0 5 10 15 20 25εa: %(m)

500


120.6150.835

6738

20.2441

20.8540.892

3323

20.2281

S4.3 S1.21000

1500

S4.3

0

σ 1 2

σ3:

kP

a S1.4

0 5 10 15 20 25εa: %(n)

500


20.7800.816

5343

20.3041

120.5510.763

7457

20.3171

S1.4 S2.6

1000

S2.6

S4.4

120.5550.767

7756

20.3131

S4.4


percentage compaction (PC � 100(1� emin)=(1� e)%), relativedensity (Dr), etc. For the latter purpose, the void ratio e of asilty sand at es � emax,HS is termed the transition void ratioetrans,SM; Dr and PC at es � emax,HS are termed the transitionrelative density Dr(trans) and transition relative percentage com-paction PC(trans), respectively. Dr(trans) or PC(trans) provides a de-marcation between case 1 (es , emax,HS, Dr . Dr(trans), PC .PC(trans)), case 2 (es near emax,HS, Dr near Dr(trans), PC nearPC(trans)), and case 3 (es . emax,HS, Dr , Dr(trans), PC , PC(trans)).Using equation (2), the transition void ratio etrans,SM is given by

etrans,SM � emax,HS 1ÿ FC

100

� �ÿ FC

100(6)

Dr(trans)(� (emax,SM ÿ etrans,SM)=(emax,SM ÿ emin,SM)] and PC(trans)

(� (1� emin,SM)=(1� etrans,SM)) are given by

Dr(trans) �[emax,SM ÿ emax,HS[1ÿ (FC=100)]� (FC=100)]

(emax,SM ÿ emin,SM)100% (7)

PC(trans) � (1� emin,SM)

f1� emax,HS[1ÿ (FC=100)]ÿ (FC=100)g 100%

(8)

where emax,SM and emin,SM are the maximum (ASTM D4254)and minimum (ASTM D1557) global void ratios of the siltysand, respectively, at a ®nes content FC. Figs 17 and 18 showthe Dr(trans) and PC(trans) versus the ®nes content data for the sixsilty sands and one clayey sand. Dr(trans) is about 40±50% forFC � 12%. It exceeds 80±90% for FC � 20±30%. It exceeds100% at a ®nes content of about 30%. PC(trans) values are about78±87% at FC � 12% and 85±100% at FC � 20±30%. Itexceeds 100% at a ®nes content of about 30%.

At low FC, less than about 20±30%, at Dr . Dr(trans) orPC . PC(trans), a silty sand behaves like the host sand at thesame es (or ø or øs). At Dr near or less than Dr(trans) (or PCnear or less than PC(trans)), the behaviour is pressure sensitive.At Dr near Dr(trans) (or PC near PC(trans)), the behaviour of twosilty sands at the same es and con®ning stress (or ø or øs) issimilar, but different from that of host sand. The host sandbehaviour is expected to be weaker and more collapsible. AtDr , Dr(trans) (or PC , PC(trans)), the behaviour of two silty sandsat the same es and con®ning stress (or ø or øs) is similar. Ahost sand specimen may not be constituted at such loose states.

For practical purposes, 30% ®nes content may be used as ademarcation point to differentiate a silty sand from a silt (case4). At that stage, the soil behaviour is primarily affected by the®nes, with a little secondary in¯uence of the coarser grains.Inter®ne void ratio and con®ning stress (or ø or øs) may beused to characterize the behaviour.

The regions corresponding to the four cases discussed beforeare shown in Figs 17 and 18.

Another way to approximately study the behaviour of siltysands is to reduce the discussions presented before in terms ofes to what is introduced as intergranular relative density Drs.Since emax,HS and emin,HS are different for different host sands.One may normalize the intergranular void ratio to obtain theintergranular relative density, de®ned as

Drs � emax,HS ÿ es

emax,HS ÿ emin,HS

(9)

In essence, the term Drs is an index of the density of thecoarser-grain structure in a silty sand normalized using themaximum and minimum void ratio of the host sand asthe frames of reference. Since it is possible to have es . emax,HS

in a silty sand, it is possible to have Drs , 0. The discussionspresented before in terms of cases 1±3 reduce to Drs . 0, Drs

near 0, and Drs , 0, respectively. Case 4 cannot be categorizedusing Drs, except for the fact that Drs , 0 can be within case 4.One may compare the behaviour of a silty sand with another interms of Drs. But the effect of con®ning stress must also betaken into consideration. One may use øs and Drs to compara-tively study the behaviour of silty sands within the limitationsdiscussed in this paper.

The discussions contained in this paper hold for freshlydeposited gap-graded silty sands. Exceptions are expected whenphysico-chemical effects, ageing, etc. are present. In the pre-sence of the latter, a silt particle may not be free to movebetween pores as hypothesized. The soil may behave as aweakly cemented sandstone. A similar limitation prevails whenthe deposition results in layering.

CONCLUDING REMARKS

A careful reanalysis of the basic concepts of critical state soilmechanics is presented. It is extended to silty sands. A new setof intergranular and inter®ne state variables (øs, es and øf , ef )are introduced to characterize the stress±strain behaviour of siltysoils. The new formulation is used as an intuitive framework tostudy the behaviour of silty sands compared with that of the hostsand. Experimental data indicating the in¯uence of intergranularvoid ratio and intergranular state parameter on the stress±strainbehaviour of silty sands are presented. Results indicate that, ingeneral, when es , emax,HS, the soil behaviour is governed by thecoarser grain contacts. The stress±strain behaviour of a silty sandspecimen at an intergranular void ratio es(, emax,HS) is similar tothat of the host sand at a void ratio e equal to es. Similar stress±strain behaviour is observed at the same ø or øs.

Even though it may be dif®cult to constitute a clean sand atstates looser than emax,HS, in the case of silty sands, it ispossible to reach intergranular void ratios larger than emax,HS.

?

?

?

100

80

60

40

20

0

Dr(

tran

s): %

Case 1

Case 2

Clayey sandCase 3

Clayey sand

0 10 20 30 40 50FC: %

Case 4

Host sand A

Ottawa C109-KC

Ottawa C109-CSF

Brenda 20/200-A

Brenda 20/200-B

Ottawa F55

Toyoura sand

Fig. 17. Transition relative density (Dr(trans)) versus ®nes content

100

95

90

85

80

75

PC

(tra

ns):

%

Case 1

Case 2

Clayey sand

Case 3

Clayey sand

0 10 20 30 40 50FC: %

Case 4

Host sand A

Ottawa C109-KC

Ottawa C109-CSF

Brenda 20/200-A

Brenda 20/200-B

Ottawa F55

Toyoura sandClayey sand

Fig. 18. Transition relative percentage compaction (PC(trans)) versus®nes content


At low ®nes content, when es . emax,HS, the stress±strainresponse of the silty sand is weak. The weak response appearsto be caused by signi®cantly reduced sand grain contacts; atsuch very loose states the stress±strain response normalizedwith respect to the initial con®ning stress is nearly the same forthe same es (or ø or øs). However, at es in the vicinity ofemax,HS, silty sands exhibit similar stress±strain behaviour, butdifferent from the host sand, at the same es and the samecon®ning stress (or at the same ø or øs).

At the same void ratio and the same con®ning stress,addition of ®nes makes the stress±strain response weaker andmore contractive. This occurs up to a threshold value in thevicinity of about 30% ®nes content. However, with furtherincrease in ®nes content, the soil is expected to become lesscontractive and stronger again. The behaviour is governedprimarily by the ®nes. The above transition ®nes contentdepends on the void ratio and the characteristics of the hostsand and the silt.

For practical ®eld applications, the matrix diagram (Fig. 3(a))can be depicted as relative density or relative percentagecompaction versus ®nes content relation, as shown in Figs 17and 18. If the ®nes content exceeds about 30%, then Dr(trans)

and PC(trans) are greater than 100%. Beyond this ®nes contentthe behaviour of that silty sand would be entirely different fromthat of the host sand; it behaves primarily as a silt with somereinforcement effect by the sand grains. If the ®nes content isless than 30%, however, a silty sand at Dr . Dr(trans) (orPC . PC(trans)) behaves like the host sand at the same intergra-nular void ratio (or the same ø or øs); if Dr , Dr(trans) (orPC , PC(trans)), the stress±strain±strength behaviour is expectedto be weak and is dependent on the initial effective con®ningstress. Two silty sands would have similar behaviour whencompared at the same es (or ø or øs). However, at Dr nearDr(trans) (or PC near PC(trans)), silty sands exhibit similar stress±strain behaviour, but different from the host sand, at the samees and the same con®ning stress (or at the same ø or øs).

ACKNOWLEDGEMENTS

zThe authors would like to thank the reviewers and theadvisory panel of GeÂotechnique for their constructive commentsand suggestions for improvement that greatly enhanced the valueof this paper. The ®nancial support for this research program bythe National Science Foundation is greatly acknowledged.

NOTATIONDr global relative density

Drs intergranular relative densityDr(trans) transition relative density

e global void ratioef inter®ne void ratio

emax,HF maximum void ratio of the host ®nesemax,HS maximum void ratio of the host sandemin,HS minimum void ratio of the host sandemax,SM maximum void ratio of the silty sandemin,SM minimum void ratio of the silty sand

es intergranular void ratioetrans transition void ratio

FC ®nes content (percentage passing through No. 200(0´074 mm) sieve) by total weight of solids

p9 mean effective stress � (ó 91 � 2ó 93)=3p9f mean effective stress at steady statePC relative percentage compaction

PC(trans) transition percentage compactionq (ó 91 ÿ ó 93)=2

Sus steady-state undrained shear strength at large axial strain(20±25%)

ó 9vo initial effective con®ning stressó 9c initial effective con®ning stressø state parameterøf inter®ne state parameterøs intergranular state parameterøms modi®ed intergranular state parameterj angle of shearing resistance at large strains

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