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1 Durham Rsarch Onlin Dpositd in DRO: 02 Jun 2008 Vrsion of attachd l: Publishd Vrsion Pr-rviw status of attachd l: Pr-rviwd Citation for publishd itm: Gallipoli, D. and Gns, A. and Sharma, R. and Vaunat, J. (2003) 'An lasto-plastic modl for unsaturatd soil incorporating th cts of suction and dgr of saturation on mchanical bhaviour.', G otchniqu., 53 (). pp Furthr information on publishr's wbsit: otchniquarticlid=36journalmnu=trujourna Publishr's copyright statmnt: Additional information: Us policy Th full-txt may b usd and/or rproducd, and givn to third partis in any format or mdium, without prior prmission or charg, for prsonal rsarch or study, ducational, or not-for-prot purposs providd that: a full bibliographic rfrnc is mad to th original sourc a link is mad to th mtadata rcord in DRO th full-txt is not changd in any way Th full-txt must not b sold in any format or mdium without th formal prmission of th copyright holdrs. Plas consult th full DRO policy for furthr dtails. Durham Univrsity Library, Stockton Road, Durham DH 3LY, Unitd Kingdom Tl : +44 (0) Fax : +44 (0)

2 Gallipoli, D., Gns, A., Sharma, R. & Vaunat, J. (2003). Géotchniqu 53, No., An lasto-plastic modl for unsaturatd soil incorporating th ffcts of suction and dgr of saturation on mchanical bhaviour D. GALLIPOLI, A. GENS, R. SHARMA{ and J. VAUNAT Th papr prsnts an lasto-plastic modl for unsaturatd soils that taks xplicitly into account th mchanisms with which suction affcts mchanical bhaviour as wll as thir dpndnc on dgr of saturation. Th proposd modl is formulatd in trms of two constitutiv variabls dirctly rlatd to ths suction mchanisms: th avrag sklton strss, which includs th avrag fluid prssur acting on th soil pors, and an additional scalar constitutiv variabl, î, rlatd to th magnitud of th bonding ffct xrtd by mniscus watr at th intr-particl contacts. Th formulation of th modl in trms of variabls closly rlatd to spcific bhaviour mchanisms lads to a rmarkabl unification of xprimntal rsults of tsts carrid out with diffrnt suctions. Th analysis of xprimntal isotropic comprssion data strongly suggsts that th quotint btwn th void ratio,, of an unsaturatd soil and th void ratio s, corrsponding to th saturatd stat at th sam avrag soil sklton strss, is a uniqu function of th bonding ffct du to watr mnisci at th intr-particl contacts. Th sam rsult is obtaind whn xamining critical stats at diffrnt suctions. Basd on ths obsrvations, an lastoplastic constitutiv modl is dvlopd using a singl yild surfac th siz of which is controlld by volumtric hardning. In spit of this simplicity, it is shown that th modl rproducs corrctly many important faturs of unsaturatd soil bhaviour. It is spcially rmarkabl that, although only on yild surfac is usd in th formulation of th modl, th irrvrsibl bhaviour in wtting drying cycls is wll capturd. Bcaus of th bhaviour normalisation achivd by th modl, th rsulting constitutiv law is conomical in trms of th numbr of tsts rquird for paramtr dtrmination. KEYWORDS: clays; constitutiv rlations; partial saturation; plasticity; suction Ct xposé présnt un modèl élasto-plastiqu pour ds sols non saturés, modèl qui prnd xplicitmnt n compt ls mécanisms par lsquls la succion affct l comportmnt mécaniqu ainsi qu lur dépndanc visà-vis du dgré d saturation. L modèl proposé st formulé n trms d dux variabls constitutivs dirctmnt liés à cs mécanisms d succion ; la contraint moynn du squltt, qui inclut la prssion d fluid moynn sur ls pors du sol t un autr variant constitutiv scalair supplémntair, î, lié à la magnitud d l fft d adhérnc xrcé par l au ménisqu aux contacts ntr particuls. La formulation du modèl n trms d variabls étroitmnt liés à ds mécanisms d comportmnt spécifiqus conduit à un unification rmarquabl ds résultats xpérimntaux ds ssais ffctués avc différnts succions. L analys ds donnés xpérimntals d comprssion isotrop suggèr fortmnt qu l quotint ntr l taux d por d un sol non saturé t l taux d por s corrspondant à l état saturé pour la mêm contraint moynn d squltt d sol, st un fonction uniqu d l fft d adhérnc dû aux ménisqus d l au aux contacts ntr particuls. L mêm résultat st obtnu lorsqu l on xamin ls états critiqus à différnts succions. En nous basant sur cs obsrvations, nous avons dévloppé un modèl constitutif élasto-plastiqu utilisant un uniqu surfac d écoulmnt dont la dimnsion st commandé par l durcissmnt volumétriqu. Nous montrons qu l modèl, malgré sa simplicité, rproduit corrctmnt bin ds caractéristiqus importants du comportmnt d un sol non saturé. Il st particulièrmnt frappant d constatr qu, bin qu un sul surfac d écoulmnt ait été utilisé dans la formulation du modèl, l comportmnt irrévrsibl dans ls cycls mouillag-séchag st bin rprésnté. En raison d la normalisation d comportmnt obtnu grâc au modèl, la loi constitutiv qui n résult a bsoin d moins d ssais pour la détrmination ds paramètrs. INTRODUCTION Suction has long bn rcognisd as a fundamntal variabl in th undrstanding of th mchanical bhaviour of unsaturatd soils. For this rasons many wll-known constitutiv modls (Alonso t al., 990; Whlr & Sivakumar, 995; Cui & Dlag, 996) includ suction as a basic strss variabl togthr with th nt strss, ó (dfind as total strss minus por air prssur). In fact suction influncs th mchanical bhaviour of an unsaturatd soil in two diffrnt ways (Karub & Kato, 994; Whlr & Karub, 995): Manuscript rcivd May 2002; rvisd manuscript accptd 20 August Discussion on this papr closs August 2003, for furthr dtails s p. ii. Dpartamnto d Ingniría dl Trrno, Univrsitat Politécnica d Catalunya, Barclona, Spain. { Univrsity of Bradford, UK (formrly Univrsity of Oxford, UK). by modifying th sklton strss through changs in th avrag fluid prssur acting in th soil pors by providing an additional bonding forc at th particl contacts, oftn attributd to capillary phnomna occurring in th watr mnisci. It is important to ralis that, for th two mchanisms, th ffcts of suction ar influncd by th stat of saturation of th soil. Th rlativ ara ovr which th watr and air prssurs act dpnds dirctly on th dgr of saturation (th prcntag of por voids occupid by watr), but th sam paramtr also affcts th numbr and intnsity of capillary-inducd intr-particl forcs. Thrfor modls using only suction in thir formulation ar unlikly to b complt. It is ncssary to incorporat, through a paramtr such as dgr of saturation, information rgarding th proportion of th soil ovr which suction ffcts ar rlvant. Thrfor it is not surprising that constitutiv modls that us only suction as th unsaturatd 23

3 24 GALLIPOLI, GENS, SHARMA AND VAUNAT variabl fac difficultis in dscribing important faturs of unsaturatd soil bhaviour. Anothr class of lasto-plastic modls for unsaturatd soils, such as thos proposd by Jommi & Di Prisco (994), Bolzon t al. (996), Lort & Khalili (2000) and Karub & Kawai (200), ar xprssd in trms of a diffrnt st of constitutiv variabls that xplicitly includ th dgr of saturation in thir dfinitions. In ths modls th strss variabl has th form of th Bishop (959) strss: ó 9 hk ¼ ó hk ä hk [u a (u a u w )] () Void ratio Scond wtting First drying First wtting whr ó 9 hk is th Bishop (959) strss, ó hk is th total strss, u a is th air prssur, u w is th watr prssur, ä hk is Kronckr s dlta, and is a soil paramtr that rangs btwn (at saturation) and zro (at dry conditions), and which is a function of dgr of saturation. Th additional scalar variabl is givn ithr by suction or by dgr of saturation dpnding on th spcific modl, with th xcption of th modl of Karub & Kawai (200), in which it is a function of both suction and dgr of saturation. Although such modls xplicitly introduc th dgr of saturation in th dfinition of th soil variabls, thy still prsnt limitations whn prdicting important aspcts of unsaturatd soil bhaviour unlss th additional complxity of multipl yild surfacs is introducd. Two xampls of faturs of bhaviour that rquir adquat modlling ar: th irrvrsibl rduction of spcific volum that can occur during drying of an unsaturatd soil (that is, during incras of suction) th dpndnc of th soil rspons during virgin loading at constant suction on th past history of suction variation. Th first typ of bhaviour has bn obsrvd by Alonso t al. (995) and Sharma (998) in laboratory tsts involving wtting drying cycls (that is, cycls of dcras incras of suction) on soil sampls subjctd to odomtric and isotropic conditions rspctivly. Exampls of this typ of soil rspons ar shown in Fig.. Th scond typ of bhaviour has bn obsrvd by Sharma (998) during isotropic loading to virgin stats of sampls at constant suction. In particular, th rsults by Sharma (998) suggst that a compactd unsaturatd soil shows a diffrnt stiffnss during virgin loading at th sam constant suction dpnding on whthr th sampl undrgos a wtting drying cycl prior to loading or not. Fig. 2 shows xampls of isotropic tsts whr th dpndnc of th soil stiffnss during virgin loading on th prvious history of suction variation can b obsrvd. Morovr, for som of ths modls th dtrmination of modl paramtrs rquirs ithr non-convntional laboratory tsts (for xampl, tsts at constant dgr of saturation) or a back-analysis for fitting modl prdictions to convntional laboratory rsults. In this papr a modl is dscribd that incorporats xplicitly in its formulation th two distinct suction ffcts mntiond prviously, including thir dpndnc on dgr of saturation. By staying clos to ths basic bhaviour mchanisms, th proposd lasto-plastic modl is capabl of rproducing th most important pattrns of unsaturatd soil mchanical bhaviour, including thos indicatd abov, in a rathr simpl mannr mploying only a singl yild surfac. Th modl also provids an ffctiv way of unifying xprimntal rsults of tsts prformd at diffrnt suctions. Apart from th concptual bnfits of such unification, this fact rsults in conomical procdurs for paramtr dtrmination from th point of viw of th numbr of laboratory tsts rquird. Volumtric strain: % Suction: kpa First wtting Scond wtting Scond drying First wtting Third wtting Scond drying Suction: kpa Fig.. Wtting drying cycl on: compactd mixtur of bntonit and kaolin undr constant isotropic nt strss of 0 kpa (aftr Sharma, 998); compactd Boom clay undr odomtric conditions at constant vrtical strss of 400 kpa (aftr Alonso t al., 995) MODELLING ASSUMPTIONS In th proposd modl th basic strss variabl is th avrag sklton strss (Jommi, 2000), which is quivalnt to th Bishop (959) strss whr th paramtr of quation () is qual to th dgr of saturation, S r : ó9 hk ¼ ó hk ä hk [u a (u a u w )] (2) This variabl xprsss th avrag strss acting in th soil sklton: that is, th diffrnc btwn th total strss and th avrag prssur of th two fluid phass (i.. gas and liquid), with th dgr of saturation as a wighting paramtr. It thrfor incorporats in a dirct mannr th first of th suction rols notd abov. Th dfinition of th avrag sklton strss rprsnts a natural xtnsion to th unsaturatd domain of th Trzaghi (936) ffctiv strss for saturatd granular matrials, and it rducs to th Trzaghi ffctiv strss at saturatd condition (that is, dgr of saturation qual to unity).

4 Void ratio Void ratio. 3.. Aftr wtting drying cycl Bfor wtting drying cycl Man nt strss: kpa Bfor wtting drying cycl Aftr wtting drying cycl Man nt strss: kpa Fig. 2. Isotropic virgin loading of compactd mixtur of bntonit and kaolin at: constant suction of 300 kpa (aftr Sharma, 998); constant suction of 200 kpa (aftr Sharma, 998) Laboratory tsts hav shown, howvr, that it is not possibl to xplain important faturs of th bhaviour of unsaturatd soils, such as th irrvrsibl comprssion (collaps) during wtting (that is, during a suction rduction) and th incras of th pr-consolidation prssur with incrasing suction, by using th avrag sklton strss as th only constitutiv variabl (.g. Jnnings & Burland, 962). To account for ths phnomna it is ncssary to considr th scond suction mchanism. Th irrvrsibl mchanical rspons of a granular matrial is associatd mainly with th rlativ slippag taking plac at th intrfac btwn soil particls. In an unsaturatd soil th possibility of such slippag is partially rducd by th stabilising ffct of th normal forc xrtd at th intr-particl contacts by mniscus lnss of watr at ngativ prssur (Whlr & Karub, 995). Svral faturs of th lasto-plastic bhaviour of unsaturatd soil ar thrfor likly to b th consqunc of ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 25 bonding and d-bonding phnomna btwn soil particls du to th formation and vanishing of watr mnisci at intr-particl contacts, and thy cannot b accountd for by using xclusivly th avrag sklton strss as a constitutiv variabl. Consquntly an additional constitutiv variabl, î, nds to b introducd as a masur of th magnitud of th intrparticl bonding du to watr mnisci so that th scond typ of suction ffct is proprly accountd for. Th magnitud of such intr-particl bonding is xpctd to b th rsult of two contributions: th numbr of watr mnisci pr unit volum of th solid fraction th intnsity of th stabilising normal forc xrtd at th intr-particl contact by a singl watr mniscus. Hnc th variabl î is dfind in th prsnt formulation as th product of two factors: th dgr of saturation of th air, ( S r ), and th function of suction, f (s): î ¼ f (s)( S r ) (3) Th factor ( S r ) accounts for th numbr of watr mnisci pr unit volum of solid fraction. Th xistnc of a uniqu rlationship btwn th valu of ( S r ) and th numbr of watr mnisci pr unit volum of solid fraction is a physically rasonabl assumption; howvr, th uniqunss of such rlationship is rigorously tru only for th idal cas whr th soil is rigid (that is, whn th dimnsions and shaps of voids do not chang as a rsult particl rarrangmnts), and whr ach valu of dgr of saturation corrsponds to a givn arrangmnt of watr within soil pors. Th trm ( S r ) is qual to zro whn th soil is saturatd (that is, S r ¼ ) and watr mnisci ar absnt, whras it assums positiv incrasing valus whn th numbr of watr mnisci incrass. Th numbr of watr mnisci pr unit volum of solid fraction can thrfor b xprssd as a monotonic incrasing function of th trm ( S r ). Th validity of this dfinition dos not apply to th cas of a soil in an xtrmly dry stat, whn th watr mnisci will start to disappar from th particl contacts. Although th xtnsion to th cas of xtrmly dry soils should not prsnt any concptual difficulty, this is not covrd in th prsnt papr bcaus th xprimntal validation would rquir xprimntal rsults from a tst programm conductd on sampls at vry low dgr of saturation, which ar currntly unavailabl. Clarly, th rlationship btwn th numbr of watr mnisci pr unit volum of solid fraction and th trm ( S r ) is dpndnt on th spcific fabric of th soil (that is, on th por siz distribution of th soil). Howvr, for th purposs of this work it is not ncssary to charactris such a rlationship xplicitly, bcaus this information is implicit in th dfinition of th function, introducd latr in th papr, that provids th variation of th ratio = s in trms of th bonding variabl î. Th function of suction f (s), which multiplis th factor ( S r ), varis monotonically btwn and 5 for valus of suction ranging btwn zro and infinity rspctivly, and it accounts for th incras with incrasing suction of th stabilising intr-particl forc xrtd by a singl mniscus. In particular, it xprsss th ratio btwn th valu of stabilising forc at a givn suction, s, and th valu of stabilising forc at a suction of zro for th idal cas of a watr mniscus locatd at th contact btwn two idntical sphrs (th analytical solution of this problm is du to Fishr, 926). Th spcific form of th function f (s) dpnds on th siz of th sphrs and on th valu of th watr surfac tnsion, but th rang of variation, btwn and 5, is always th sam rgardlss of dimnsions and

5 26 GALLIPOLI, GENS, SHARMA AND VAUNAT physical proprtis. Th rlationship f (s) usd in this work is shown in Fig. 3, corrsponding to th cas of two sphrs having radii of ìm and a valu of th surfac tnsion of watr corrsponding to a tmpratur of 208C. Hains (925) suggstd that a matrial with th txtur of a compactd kaolin could b rprsntd by sphrs having radii qual to ìm. Obviously th shaps of th aggrgats ar far from bing sphrs of th sam siz. In addition, for soils with a multi-modal por siz distribution, th dimnsion of th sphrical grains in th solution of Fishr (926) should b dfind as a variabl dpnding on th avrag siz of th soil pors that includ watr mnisci. At this stag of dvlopmnt of th modl, howvr, th assumption of a simplifid rlationship, such as th on givn in Fig. 3, is considrd rasonabl. Th prsnc of mniscus watr provids a physical xplanation for th xprimntal obsrvation that, at th sam valu of avrag sklton strss, th valu of void ratio during virgin loading of unsaturatd soil is always gratr than th valu for th sam soil subjctd to th sam load undr saturatd conditions. Th xistnc of watr in th form of mniscus lnss within an unsaturatd soil maks th intr-particl contacts mor stabl, and thrfor rstrains th rciprocal slippag of soil particls that causs comprssiv strains during virgin loading. Consistnt with such mpirical obsrvations, this work introducs a fundamntal modlling assumption spcifying that, during virgin loading of an unsaturatd soil, th ratio = s btwn void ratio in unsaturatd conditions,, and void ratio in saturatd conditions, s, at th sam avrag sklton strss stat is a uniqu function of th bonding variabl, î. This assumption not only provids an ssntial starting point for th dvlopmnt of th modl, it also offrs a powrful unifying prspctiv to xamin th rsults of tsts prformd at diffrnt suctions. This assumption is validatd in th nxt sction on th basis of publishd laboratory tst data. EXPERIMENTAL VALIDATION OF MODELLING ASSUMPTIONS Th validation of th assumption introducd in th prvious sction has involvd th analysis of diffrnt sts of f(s) s: kpa Fig. 3. Ratio btwn intr-particl forcs at suction s and at null suction du to a watr mniscus locatd at th contact btwn two idntical sphrs (analytical solution by Fishr, 926) data from laboratory tsts prformd on compactd Spswhit kaolin (Sivakumar, 993; Whlr & Sivakumar, 2000), on a compactd mixtur of bntonit and kaolin (Sharma, 998), and on compactd Kiunyu gravl (Toll, 990). Th first part of this sction analyss th data from isotropic virgin comprssion tsts at constant suction (Sivakumar, 993; Sharma, 998). At th nd of th sction, th analysis of furthr xprimntal data from triaxial shar tsts on compactd Spswhit kaolin (Sivakumar, 993; Whlr & Sivakumar, 2000) and on compactd Kiunyu gravl (Toll, 990) dmonstrats that th conclusion achivd for isotropic strss stats can also b xtndd to non-isotropic strss stats. Sivakumar (993) and Sharma (998) prformd isotropic virgin comprssion of soil sampls at diffrnt valus of suction 00 kpa, 200 kpa and 300 kpa as wll as of saturatd sampls. During ths tsts th corrsponding changs of void ratio,, and watr ratio, w (that is, th volum of watr in a volum of soil containing unit volum of solids), wr masurd. Th analysis of th xprimntal rsults indicats that, for th rang of strsss considrd, th normal comprssion lins at constant suction follow a linar rlationship in th smi-logarithmic plans ln p and w ln p (whr p is th isotropic nt strss). Each normal comprssion lin is thrfor idntifid by th valus of th two paramtrs that corrspond to th slop and to th intrcpt at a givn valu of p. As for th data st by Sharma (998) thr wr no virgin loading tsts on saturatd sampls; th slop and intrcpt of th saturatd normal comprssion lin wr stimatd from th drying branch of a wtting drying tst undr isotropic constant load. In this typ of tst, aftr an initial wtting that brought th soil to saturation, th sampl was subjctd to drying that causd significant irrvrsibl changs of void ratio. Th principl of ffctiv strss holds during most of such drying bcaus th sampl rmaind saturatd for a larg incras of suction owing to its high air-ntry valu. Undr saturatd conditions th imposd chang of suction corrsponds to an quivalnt chang of th ffctiv strss. Hnc, by plotting th void ratio against th isotropic ffctiv strss, it was possibl to stimat th slop and th intrcpt of th saturatd normal comprssion lin. Th valus of slops and intrcpts of normal comprssion lins of and w at constant suction wr usd to r-plot th normal comprssion lins in trms of th isotropic avrag sklton strss, p0. Figs 4 and 5 show th normal comprssion lins at constant suctions of zro (saturatd), 00 kpa, 200 kpa and 300 kpa, in th smi-logarithmic plan ln p0 for ach st of data rspctivly. Inspction of Figs 4 and 5 rvals that th normal comprssion lins at non-zro valus of suction ar not straight lins in th smi-logarithmic plan ln p0, but thy ar curvs with slops that dcras as thy approach th saturatd lin (zro suction). This is consistnt with th xprimntal obsrvation that th dgr of saturation incrass during isotropic loading to virgin stats at constant suction. Indd, if a soil sampl attains saturation during comprssion at a positiv valu of suction, th isotropic avrag sklton strss coincids with th saturatd ffctiv strss, and th corrsponding valu of void ratio should li on th saturatd normal comprssion lin. Aftr saturation, th normal comprssion lin at non-zro suction should thrfor hav th sam slop as th saturatd normal comprssion lin. It is thrfor to b xpctd that th slop of th normal comprssion lins at non-zro suction progrssivly rducs as thy convrg towards th saturatd lin. From th normal comprssion lins in th smi-logarithmic plan ln p0 shown in Figs 4 and 5 it is possibl to calculat th ratio btwn th valu of th unsaturatd

6 ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL Modl quation Exprimntal (s 00 kpa) Exprimntal (s 200 kpa) Exprimntal (s 300 kpa). / s 0. 8 s 0 kpa (saturatd) s 00 kpa s 200 kpa s 300 kpa Fig. 4. Normal comprssion lins at constant suction in th plan ln p0 (data by Sharma, 998) Fig. 6. Rlationship btwn ratio = s and bonding factor î during isotropic virgin loading at constant suction (data by Sharma, 998) Modl quation Exprimntal (s 00 kpa) Exprimntal (s 200 kpa) Exprimntal (s 300 kpa). / s 0. 8 s 0 kpa (saturatd) s 00 kpa s 200 kpa s 300 kpa Fig. 5. Normal comprssion lins at constant suction in th plan ln p0 (data by Sivakumar, 993) Fig. 7. Rlationship btwn ratio = s and bonding factor î during isotropic virgin loading at constant suction (data by Sivakumar, 993) 0. 6 soil and that corrsponding to th saturatd stat, s, at th sam avrag sklton strss. Figs 6 and 7 show, for th data sts of Sivakumar (993) and Sharma (998) rspctivly, th valu of th ratio = s plottd against th valu of th bonding variabl, î, dfind by quation (3) (corrsponding to th valus of S r and f (s) of th unsaturatd soil). Th valu of th function of suction, f (s), has bn calculatd according to th rlationship shown in Fig. 3, and it is qual to 0, 5 and 8 for suction valus of 00 kpa, 200 kpa and 300 kpa rspctivly. Th rlationship shown in Fig. 3 rfrs to th Fishr (926) solution whr th sphrical grains hav radii qual to ìm. This is th ordr of magnitud of th macrostructural voids of a clay soil compactd dry of optimum, such as th soils invstigatd by Sivakumar (993) and Sharma (998). Porosimtry studis hav shown that clay matrials compactd dry of optimum prsnt a markd bimodal por siz distribution (.g. Gns & Alonso, 992) with macrostructural and microstructural voids of th ordr of magnitud of ìm and 0 0 ìm rspctivly. For th rang of suctions invstigatd by Sivakumar (993) and Sharma (998) it is rasonabl to xpct that only macro-voids ar affctd by dsaturation (and hnc by th formation of watr mnisci) whil th micro-voids stay saturatd. Inspction of Figs 6 and 7 suggsts rmarkably that, for all th thr valus of suction invstigatd, th data from normal comprssion ar consistnt with a uniqu rlationship linking th valu of th proportion = s and th bonding variabl, î. Such a bonding variabl thrfor appars to b uniquly rlatd to th ability of th sklton to sustain

7 28 GALLIPOLI, GENS, SHARMA AND VAUNAT highr void ratios whn th soil is undr suction. For ach of th thr curvs at constant suction shown in Figs 6 and 7, th valu of th proportion = s is xpctd to attain a valu of whn î is qual to zro (that is, whn th sampl achivs saturation) bcaus in this cas th normal comprssion lins at non-zro suctions coincid with th saturatd lin in th smi-logarithmic plan ln p0. Th modl quation that fits th thr curvs of = s against î at constant suction in Figs 6 and 7 has th following form: ¼ a [ xp (b î)] (4) s whr a and b ar fitting paramtrs. Equation (4) prdicts a valu of = s qual to whn î is qual to zro, consistnt with th physical xplanation givn abov. For th rang of suction invstigatd, th valu of th function f (s) varis rlativly littl in comparison with th variation of th valu of th bonding variabl î. It is thrfor rasonabl to xpct that xprimntal data might b similarly consistnt with a rlationship linking th valu of th proportion = s during isotropic virgin loading to th valu of th dgr of saturation of th gas phas, ( S r ). Sivakumar (993) and Whlr & Sivakumar (2000) prsntd furthr xprimntal data from sharing tsts to critical stat on compactd Spswhit kaolin undr various suction valus. Ths data hav bn usd to invstigat whthr th rlationship btwn th ratio = s and th bonding variabl î could b xtndd to non-isotropic strss stats. Th modl quation in Fig. 7, which had bn dfind on th basis of isotropic normal comprssion tsts, was thrfor usd to prdict valus of void ratio at critical stats. Th prdictd valus of void ratio at critical stat wr computd following th sam procdur as in th isotropic cas. First th saturatd critical-stat lin in th smi-logarithmic plan ( ln p0) was dfind (by fixing its slop and intrcpt) on th basis of sharing tsts prformd by Sivakumar (993) and Whlr & Sivakumar (2000) on saturatd sampls. Thn, for ach unsaturatd sampl shard to critical stat, th corrsponding xprimntal valus of isotropic nt strss, dgr of saturation and suction at critical stat wr usd to calculat th isotropic avrag sklton strss, p0, and th bonding variabl î. Ths valus of p0 and î wr thn mployd to comput th void ratio, s, from th saturatd critical-stat lin and th ratio = s from th modl quation of Fig. 7 rspctivly. Fig. 8 shows th comparison btwn prdictd and xprimntal valus of void ratio at critical stat corrsponding to diffrnt suction lvls. Inspction of Fig. 8 indicats rmarkably that th rlationship stablishd btwn th ratio = s and th bonding factor î for isotropic virgin comprssion and givn by quation (4) can also accuratly prdict th void ratio valus at critical stat. This implis that such a rlationship might b uniqu for th lasto-plastic loading of an unsaturatd soil rgardlss of th spcific strss ratio applid to th sampl, and that th slctd bonding variabl closly rprsnts th ral ffct of suction on intr-granular strss. Th diffrnt sris of Fig. 8 corrspond to diffrnt procdurs adoptd by Whlr & Sivakumar (2000) for th compaction of Spswhit kaolin at th sam dry of optimum watr contnt (that is, for th tst sris II and III th compaction prssur and mthod of compaction wr diffrnt from thos mployd for prparing th sampls from sris I shown in Figs 5 and 7). On th basis of thir mpirical rsults, Whlr & Sivakumar (2000) concludd that th bhaviour at critical stat of a soil compactd at th sam dry of optimum watr contnt is not affctd by th procdur adoptd for compaction. This rsult is also confirmd by th comparison shown in Fig. 8. Prdictd Idntity function s 00 kpa (sris I) s 200 kpa (sris I) s 300 kpa (sris I) s 00 kpa (sris II) s 300 kpa (sris II) s 00 kpa (sris III) s 300 kpa (sris III) Exprimntal Fig. 8. Comparison btwn xprimntal and prdictd void ratio at critical stat (data by Sivakumar, 993 and Whlr & Sivakumar, 2000) Finally, th analysis of th data from undraind (with rspct to th watr phas) triaxial shar tsts prformd by Toll (990) on compactd sampls of a latritic gravl from Knya (Kiunyu gravl) is prsntd. Sampls wr compactd by Toll (990) at diffrnt valus of watr contnt ranging from 7 0 to 27 7 and thn shard in axial comprssion to critical stat whil prvnting th flow of watr. Th author rports th masurd valus of th void ratio, dgr of saturation, suction and nt strss stat at th nd of th tsts whn th unsaturatd sampls hav attaind critical stat conditions. Togthr with th rsults from unsaturatd soil sampls, th author prsnts a smallr st of data from undraind triaxial shar tsts to critical stat prformd on saturatd sampls. For th saturatd tsts, th soil sampls wr compactd at watr contnt ranging btwn 8 7 and 3 0 and wr thn saturatd prior to tsting. On th basis of th saturatd shar tsts, Toll (990) suggsts th valus of th slop and intrcpt of saturatd critical-stat lin in th smi-logarithmic plan ( ln p0). Th data prsntd by Toll (990) ar limitd to th critical-stat valus masurd at th nd of th undraind sharing of ach sampl. Such data wr usd in this work to validat th proposd assumption of a uniqu rlationship btwn th ratio = s (corrsponding to a givn valu of th isotropic avrag sklton strss) and th bonding variabl î at th critical stat. Th soil tstd by Toll (990) is likly to xhibit a grading diffrnt from that of th soils invstigatd by Sivakumar (993) and Sharma (998). Howvr, owing to th absnc of prcis information on th fabric of th soil tstd by Toll (990), th radii of th sphrs in th Fishr (926) solution (that is, in th function f (s) of quation (3)) wr takn qual to ìm, th sam valu as in th prvious two analyss. For ach xprimntal data point th valu of th function f (s) was thn calculatd according to th suction masurd by Toll (990) at th critical stat. For th purposs of th study prsntd hr, th unsaturatd sampls tstd by Toll (990) wr classifid in two diffrnt groups, ach on including sampls compactd at similar valus of watr contnt. Th data shown in Fig. 9 rfr to unsaturatd sampls whos compaction watr con-

8 / s. 3. Exprimntal Modl quation ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 29 suction, nt strss stat and dgr of saturation at critical stat) follow a uniqu trnd whn plottd in th plan (= s, î). In Figs 9 and 0 th xprssion of th curv intrpolating th xprimntal data is givn by quation (4) (th sam xprssion as usd for th xprimntal data of Sharma (998) and Sivakumar (993) shown in Figs 6 and 7). Th sampls compactd at watr contnts btwn 24 9% and 27 7% (Fig. 9) had suction valus ranging from 2 kpa to 73 kpa on raching th critical stat, whras th suction valus at critical stat for th sampls compactd at watr contnts btwn 9 6% and 2 9% (Fig. 0) varid btwn 22 kpa and 537 kpa. Th proposd rlationship, thrfor, is shown to b capabl of bringing togthr rsults from a vry wid rang of suction valus Fig. 9. Rlationship btwn ratio = s and bonding factor î at critical stat for soil sampls compactd at watr contnt btwn 24 9% and 27 7%. Th suctions at critical stat rang from 2 kpa to 73 kpa (data by Toll, 990) tnt rangd btwn 24 9% and 27 7%, whras for th sampls shown in Fig. 0 th compaction watr contnt rangd btwn 9 6% and 2 9%. Th valu of th compaction watr contnt significantly affcts th nsuing fabric of th unsaturatd sampl, and this in turn has an ffct on th mchanical rspons of th sampl, including th rspons at critical stat (as discussd by Whlr & Sivakumar, 2000). Hnc th purpos of such a distinction was to slct two homognous groups of sampls that prsntd a similar soil fabric, and whos mchanical rspons during laboratory tsting could thrfor b compard. From xamination of Figs 9 and 0 it can b obsrvd that all critical-stat data points (ach of thm corrsponding to diffrnt valus of / s Exprimntal Modl quation Fig. 0. Rlationship btwn ratio = s and bonding factor î at critical stat for soil sampls compactd at watr contnt btwn 9 6% and 2 9%. Th suctions at critical stat rang from 22 kpa to 537 kpa (data by Toll, 990) FORMULATION OF THE ELASTO-PLASTIC STRESS STRAIN MODEL An lasto-plastic isotropic strss strain modl for unsaturatd soils incorporating volumtric hardning is dscribd in this sction. Th succss of th modlling idas proposd hr will b dmonstratd in th nxt sction by comparing th prdictions with th xprimntal rsults from various typs of laboratory tst, all prformd undr isotropic loading. Th dvlopmnt of th modl is hnc limitd in this sction to isotropic strss stats in ordr to concntrat attntion on th basic faturs of th modl. Extnsion to mor gnral strss stats is quit straightforward following standard procdurs (Gns, 995). Th formulation of a constitutiv modl including volumtric hardning rquirs th dfinition of: a normal comprssion stat surfac, which rlats th valus of void ratio,, isotropic avrag sklton strss, p0, and bonding variabl, î, during th irrvrsibl bhaviour of th soil an incrmntal xprssion that rlats th lastic part of th chang of void ratio,, to th changs of th isotropic avrag sklton strss, p0, and bonding variabl, î. Th normal comprssion stat surfac is dfind hr as th product of two factors. Th first factor is th quation of th saturatd normal comprssion lin rlating th variation of th void ratio, s, to th chang of th isotropic avrag sklton strss, p0, and th scond factor is th quation that links th variation of th ratio = s to th chang of th bonding variabl, î. For th matrials studid hr, th analytical form of th normal comprssion stat surfac is thrfor xprssd as ( p0, î) ¼ s (î) s ( p0) (5) whr ( p0, î) is th normal comprssion stat surfac, (= s )(î) is givn by quation (4), and s ( p0) is th saturatd normal comprssion lin (a straight lin in th smilogarithmic plan ln p0) having th form s ( p0) ¼ N º ln p0 (6) N and º in quation (6) ar th intrcpt (at p0 ¼ kpa) and th slop of th saturatd normal comprssion lin rspctivly. Not that, for saturatd conditions, th isotropic avrag sklton strss, p0, coincids with th isotropic ffctiv strss, p, and thrfor th paramtrs N and º ar qual to thos that idntify th saturatd normal comprssion lin in th smi-logarithmic plan ln p. Fig. shows thr xampls of normal comprssion lins that li on th normal comprssion stat surfac and corrspond to constant valus of th bonding variabl, î. Equations (5) and (6)

9 30 GALLIPOLI, GENS, SHARMA AND VAUNAT p 0 (0) Normal comprssion lins Elastic strss path p Initial yild locus Expandd yild locus 2 p 0 ( 2 ) 2 indpndnt of th variation of th bonding variabl î). This is quivalnt to assuming that th lastic dformation of th soil sklton is not affctd by th bonding action that th watr mnisci xrt at th intr-particl contacts. It will b shown in th nxt sction that this assumption fits wll th lastic bhaviour of th laboratory tsts considrd in this work. Th normal comprssion stat surfac dfind by quation (5) acts as a limiting surfac in (, p0, î) spac, whr it sparats th rgion of attainabl soil stats from th rgion of non-attainabl soil stats. Th soil rspons is lastic whil th soil follows a path insid th spac of attainabl soil stats. Whn th soil path rachs th normal comprssion stat surfac, this surfac imposs a constraint on furthr changs of, p0 and î, and th soil stat can thrfor ithr mov back insid th spac of th attainabl soil stats or follow a path lying on th normal comprssion stat surfac. Whn th lattr possibility occurs, irrvrsibl (lasto-plastic) changs of void ratio dvlop. Thus th normal comprssion stat surfac and th lastic law introducd abov (quations (5) and (7) rspctivly) implicitly dfin a yild locus that incorporats a volumtric hardning rul. To obtain th analytical form of such a yild locus considr th lastic strss path in Fig. starting from th soil stat dnotd by, at an isotropic avrag sklton strss p o 0(0) on th saturatd normal comprssion lin, and moving to th soil stat dnotd by 2, at an isotropic avrag sklton strss p o 0(î 2 ) on th unsaturatd normal comprssion lin corrsponding to î ¼ î 2. Th chang of void ratio during th path from stat to stat 2 is computd according to th lastic quation (7): ¼ kln p o(î 0 ) (8) p o 0(0) 0 p 0 (0) indicat that th normal comprssion lins at constant î ar straight lins in th smi-logarithmic plan ln p0 and that thir xtrapolations at high valus of p0 intrsct ach othr at th sam point, coinciding with a valu of th void ratio s on th saturatd normal comprssion lin qual to zro. Th lastic chang of void ratio,, is assumd to b givn by ¼ kln p f 0 (7) p9 i p p 0 ( 2 ) Fig.. Drivation of th yild locus in th isotropic plan: chang of void ratio; strss path whr k is th lastic swlling indx and p i 0 to p f 0 ar th initial and final valu of th isotropic avrag soil sklton strss rspctivly. Equation (7) implis that th lastic chang of void ratio dpnds xclusivly on th chang of th isotropic avrag sklton strss, p0 (that is, it is As th soil stats and 2 also blong to th normal comprssion stat surfac thy must li on th sam yild locus, and an altrnativ xprssion for th variation of void ratio during th path from stat to stat 2 can thrfor b obtaind by using th normal consolidation stat surfac of quation (5): ¼ [ p0(0), 0] [ p0(î ), î ] ¼ N º ln p o 0(0) s (î )[N º ln p 0 o (î )] (9) By quating quation (8) and quation (9) and thn rarranging, th following quation of th yild locus in th isotropic plan î ln p0 is obtaind: ln p o 0(î ) ¼ º k (î) ( þ N) s ln p o 0(0) þ (î)º k (î)º k s s (0) Figur shows th yild locus of quation (0) in th isotropic plan î ln p0 togthr with th two yild points corrsponding to th soil stats and 2, which ar idntifid by th coordinats ( p 0 o (0), 0) and ( p 0 o (î 2 ), î 2 ) rspctivly. Figur also shows an xpandd yild locus, indicatd by th brokn lin, which rfrs to a soil sampl that has xprincd additional plastic volumtric strains and whos yild locus has thrfor undrgon volumtric hardning. Th currnt siz of th yild locus is idntifid by th valu of its intrcpt p 0 o (0) with th horizontal axis, which is th yild valu of th isotropic avrag sklton strss during isotropic comprssion of a saturatd sampl. Th saturatd yild strss p 0 o (0) can thrfor b assumd as th

10 hardning paramtr of th prsnt lasto-plastic modl. Th irrvrsibl chang of void ratio, p, associatd with th xpansion of th yild locus from an initial position idntifid by p o 0(0) ¼ p o 0(0) i to a final position idntifid by p o 0(0) ¼ p o 0(0) f coincids thn with th irrvrsibl chang of void ratio calculatd by th saturatd normal comprssion lin for a variation of th isotropic avrag sklton strss from p o 0(0) i to p o 0(0) f : p ¼ (º k)ln p 0 o(0) f p 0 o (0) i () and quation () thus rprsnts th volumtric hardning rul of th proposd lasto-plastic modl. Th complt modl includs a formulation to comput th dgr of saturation that must incorporat th ffct of hydraulic hystrsis and strss-inducd changs of soil fabric. Th rlationships proposd by Vaunat t al. (2000) and Gallipoli t al. (2003) can b usd for this purpos, but a dtaild dscription of this componnt of th modl is outsid th scop of th papr. For th modl computations prsntd in th nxt sction, th xprimntally obsrvd dgrs of saturation hav bn usd. In this way th diffrncs btwn prdictions and obsrvations must b attributd xclusivly to th mchanical lasto-plastic modl. MODEL PREDICTIONS Th good prformanc of th proposd lasto-plastic modl is dmonstratd hr by comparing th rsults from a slction of xprimnts prformd by Sharma (998) on a compactd mixtur of bntonit and kaolin with th corrsponding modl prdictions. In particular, th comparison will show th potntial of th proposd modl for corrctly prdicting: Th initial yild locus of th soil corrsponding to th aftr-compaction stat Th irrvrsibl chang of void ratio occurring during wtting (collaps) (c) Th irrvrsibl chang of void ratio during drying (d) th dpndnc of th soil rspons during isotropic virgin loading at constant suction on th prvious history of suction variation. Points (c) and (d) rfr to typical faturs of unsaturatd soil bhaviour that ar not takn into account by xisting lastoplastic constitutiv framworks formulatd in trms of a singl yild surfac. Th slction procss of th modl paramtr valus usd for th prdictions (s Tabl ) has bn dscribd arlir, xcpt for th valu of th lastic swlling indx, k, which was slctd on th basis of lastic isotropic loading unloading cycls at constant suction. Figurs 2 4 show th comparison btwn xprimntal and prdictd bhaviour for thr isotropic loading tsts at constant suction (00 kpa, 200 kpa and 300 kpa rspctivly) that involv lasto-plastic yilding. Inspction of Figs 2 4 rvals that th proposd modl corrctly calculats th Tabl. Paramtr valus for th proposd lasto-plastic modl Paramtr Valu º, slop of NCL at s ¼ N, on NCL at s ¼ 0 for p0 ¼ kpa 759 k, swlling indx for changs of p a, paramtr of modl quation (4) b, paramtr of modl quation (4) 49 ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL Exprimntal Prdiction p: kpa Initial yild locus Expandd yild locus Fig. 2. Modl prdiction for isotropic virgin loading at constant suction of 00 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path rspctiv yild points by assuming for all thr tst simulations th sam initial yild locus associatd with a valu of th hardning paramtr, p o 0(0) ¼ 7 kpa. Such a modl prdiction is corroboratd by th soil rspons obsrvd by Sharma (998) during th qualisation stag prior to loading, whn th suction of th thr sampls was dcrasd from th valu aftr compaction to 00 kpa, 200 kpa and 300 kpa rspctivly. During this stag all thr sampls xprincd xclusivly lastic swlling, which indicats that th initial yild curv aftr compaction had not undrgon furthr xpansion associatd with plastic volumtric comprssion (collaps). It is thn xpctd that all thr sampls would yild on th sam locus during isotropic loading, and th proposd modl indd corrctly prdicts this. Thrfor, for th tst simulations prsntd in th rmaindr of this sction, th valu of th hardning paramtr corrsponding to th soil aftr compaction was assumd to b qual to 7 kpa.

11 32 GALLIPOLI, GENS, SHARMA AND VAUNAT. 3 First loading. 3 Unloading rloading Exprimntal Prdiction Scond loading p: kpa Initial yild locus Expandd yild locus First loading Unloading rloading Exprimntal Prdiction p: kpa Initial yild locus Expandd yild locus Scond loading Fig. 3. Modl prdiction for isotropic virgin loading at constant suction of 200 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path Fig. 4. Modl prdiction for isotropic virgin loading at constant suction of 300 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path Figur 3 also shows th comparison btwn modl prdictions and xprimntal rsults for an lastic loading unloading cycl, which confirms th adquacy of th lastic law givn by quation (7). Th vry good match btwn xprimntal and prdictd valus of void ratio at th bginning of th loading in Figs 2 4 provids a furthr proof of th validity of quation (7) bcaus th prdictd initial valus of void ratio ar computd by mans of an lastic path starting from th saturatd yild strss stat ( p o 0(0) ¼ 7 kpa) according to th following xprssion: p i 0 i ¼ o k ln (2) 7 kpa whr o is th void ratio prdictd by th saturatd normal comprssion lin for p0 ¼ 7 kpa and p i 0 is th isotropic avrag sklton strss of th unsaturatd sampl at th bginning of loading. Figur 5 shows th comparison btwn xprimntal and prdictd bhaviour for a wtting drying cycl prformd at a constant isotropic nt strss of 50 kpa. Inspction of th strss path followd by th soil during th tst in Fig. 5 rvals that th modl prdicts irrvrsibl changs of void ratio during both th wtting and th drying branch of th tst. Yilding of th soil occurs initially during wtting, and th consqunt dvlopmnt of lasto-plastic strains producs th first xpansion of th yild locus from its initial position to th position indicatd by (corrsponding to th nd of wtting). Aftr th rvrsal of suction th modl continus to prdict lasto-plastic dformations during th whol drying, and this corrsponds to a furthr xpansion of th yild locus from position to th final position. This tst simulation clarly dmonstrats th potntial of th prsnt framwork to intrprt th lastoplastic volumtric strains that occur during both th wtting and th drying phass as a singl mchanical phnomnon that can b modlld by mploying only on yild locus. Part of th discrpancy btwn xprimntal rsults and modl prdiction in Fig. 5 is du to th incomplt

12 ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL Wtting First wtting Drying First drying Scond wtting. Exprimntal Prdiction Exprimntal Prdiction Scond drying s: kpa Initial yild locus Expandd yild locus Wtting s: kpa Initial yild locus Expandd yild locus First wtting 0. 2 Drying Scond wtting 0. 2 First drying 0. Scond drying Fig. 5. Modl prdiction for a wtting drying cycl of a sampl subjctd to a constant isotropic nt strss of 50 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path qualisation of suction within th sampl during th tst. Th occurrnc of incomplt qualisation is provn by th xprimntal obsrvation, rportd by Sharma (998), that a significant amount of watr flowd into th sampl during th stabilisation priod at th nd of th wtting stag, whn th sampl was kpt at constant suction of 00 kpa for a priod of tim ncssary to qualis suction bfor subsqunt drying (th vrtical chang of void ratio at constant suction of 00 kpa shown in Fig. 5 corrsponds to this stabilisation priod). Howvr, dspit such xprimntal limitations, inspction of Fig. 5 still indicats a satisfactory agrmnt btwn prdictd and computd rsults. Now th mor complx strss paths involving wtting drying cycls ar considrd. Fig. 6 shows th comparison btwn th xprimntal and prdictd bhaviour during wtting drying cycls prformd at a constant isotropic nt strss of 0 kpa. Inspction of Fig. 6 indicats that lasto-plastic strains occur xclusivly during th drying phass whras lastic swlling taks plac during th Fig. 6. Modl prdiction for wtting drying cycls of a sampl subjctd to a constant isotropic nt strss of 0 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path wtting phass. Th irrvrsibl strains gnratd by th first drying produc an xpansion of th yild locus from th initial position to position whras th scond drying originats a furthr xpansion from position to position. Not that, as xplaind abov, th discontinuity in th slop of th wtting paths shown in Fig. 6 is du to th stabilisation phas following incomplt suction qualisation during prvious wtting. Th modl prdictions in Figs 5 and 6 rprsnt a significant improvmnt ovr xisting lasto-plastic modls basd on a singl yild locus, which would incorrctly prdict lastic comprssion during all th drying phass of th abov tsts. Finally, Fig. 7 shows th comparison btwn xprimntal and prdictd bhaviour for two constant suction isotropic tsts that show diffrnt mchanical rsponss dpnding on whthr or not th sampl has undrgon a wtting drying cycl prior to loading. Inspction of Fig. 7 rvals that th modl is capabl of capturing th diffrnt stiffnsss shown by th soil during virgin loading in th two

13 34 GALLIPOLI, GENS, SHARMA AND VAUNAT Exprimntal Prdiction Loading aftr wtting drying Loading without wtting drying p: kpa Initial yild locus Expandd yild locus Wtting cass. This is a significant advanc with rspct to xisting modls that would instad prdict th sam slop of th normal comprssion lins for both cass rgardlss of th prvious history of suction variation. A furthr improvmnt with rspct to xisting modls is that th prsnt framwork corrctly prdicts diffrnt valus of void ratio at th bginning of loading in th two cass of Fig. 5. This is du to th irrvrsibl chang of void ratio that occurs during drying of th sampl subjctd to th wtting drying cycl (s Fig. 5), which is also rflctd in th diffrnt dgr of xpansion of th yild locus achivd at th nd of tst in th two cass (position and position rspctivly). Loading without wtting drying Drying Loading aftr wtting drying Fig. 7. Modl prdiction for isotropic virgin loading at constant suction of 300 kpa (xprimntal data by Sharma, 998): chang of void ratio; strss path CONCLUSIONS Th papr proposs an innovativ constitutiv framwork for unsaturatd soil that is abl to xplain th various mchanical faturs of this matrial by rsorting to a physical dscription of th diffrnt ffcts of suction on soil straining. In th assumd mchanism, th rlativ slippag of soil particls is govrnd by two countracting actions xrtd on th assmblag of soil particls: th prturbing action of th avrag strss stat acting on th soil sklton th stabilising action of th normal forc xrtd at th intr-particl contacts by watr mnisci. Th variabls controlling ach on of ths actions (that is, th avrag sklton strss variabl, ó 0, and th bonding variabl, î, rspctivly) ar dfind on th basis of th currnt valus of th nt strss stat, suction and dgr of saturation. Th introduction of dgr of saturation in th dfinition of th soil constitutiv variabls is ssntial to rprsnt proprly th contribution of soil suction to th two ffcts dscribd abov. Basd on a physical argumnt, th prsnt proposal assums that, during th lasto-plastic loading of a soil lmnt, th proportion = s btwn th void ratio,, undr unsaturatd conditions and th void ratio, s, undr saturatd conditions at th sam avrag sklton strss stat is a uniqu function of th bonding variabl, î. This fundamntal assumption is succssfully validatd in this work by th analysis of svral publishd sts of xprimntal data for diffrnt matrials. Th analysis of on st of data (for which both isotropic and sharing tsts ar availabl) also suggsts that th rlationship btwn = s and î is uniqu for a givn soil, and that it is indpndnt of th applid strss ratio. On th basis of this assumption a full lasto-plastic strss strain modl for isotropic strss stats is formulatd, and its good prformanc is dmonstratd by th comparison btwn prdictd and laboratory tsts rsults from a comprhnsiv xprimntal programm including a wid varity of diffrnt strss paths. This comparison confirms th potntial of th proposd modl for corrctly prdicting th most important faturs of th mchanical bhaviour of unsaturatd soils by rtaining at th sam tim th simplicity of a modl formulatd in trms of a singl yild curv. In particular it is abl to prdict corrctly th following two typical rsponss of unsaturatd soils that ar not modlld by xisting lasto-plastic constitutiv framworks basd on a singl yild surfac: th irrvrsibl chang of void ratio during drying th dpndnc of th rspons during virgin comprssion at constant suction on th prvious history of suction variation. An additional significant advantag is that a rducd numbr of laboratory tsts ar ncssary for calibrating th proposd modl. In particular, th rlationship btwn = s and î is th only additional information rquird for th unsaturatd soil bhaviour (apart from th paramtr valus for th saturatd modl). To dfin th rlationship btwn = s and î, it is possibl to choos among altrnativ tsting options that involv irrvrsibl straining of th soil such as virgin loading at constant suction, undraind virgin loading and wtting drying at constant applid strss. ACKNOWLEDGEMENTS Th authors wish to acknowldg th support of th Europan Commission via a Mari Curi Fllowship awardd to Dr Domnico Gallipoli and of th Ministrio d Cincia y Tcnología through rsarch grant BTE Dr Radhy Sharma carrid out th xprimntal work