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Gallipoli, D. and Gens, A. and Sharma, R. and Vaunat, J. (2003) ’An elasto-plastic model for unsaturated soilincorporating the effects of suction and degree of saturation on mechanical behaviour.’, Gotechnique., 53 (1).pp. 123-136.

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123

Gallipoli, D., Gens, A., Sharma, R. & Vaunat, J. (2003). Geotechnique 53, No. 1, 123–135

An elasto-plastic model for unsaturated soil incorporating the effects ofsuction and degree of saturation on mechanical behaviour

D. GALLIPOLI ,� A. GENS,� R. SHARMA{ and J. VAUNAT�

The paper presents an elasto-plastic model for unsatur-ated soils that takes explicitly into account the mechan-isms with which suction affects mechanical behaviour aswell as their dependence on degree of saturation. Theproposed model is formulated in terms of two constitutivevariables directly related to these suction mechanisms:the average skeleton stress, which includes the averagefluid pressure acting on the soil pores, and an additionalscalar constitutive variable, �, related to the magnitudeof the bonding effect exerted by meniscus water at theinter-particle contacts. The formulation of the model interms of variables closely related to specific behaviourmechanisms leads to a remarkable unification of experi-mental results of tests carried out with different suctions.The analysis of experimental isotropic compression datastrongly suggests that the quotient between the void ratio,e, of an unsaturated soil and the void ratio es, corre-sponding to the saturated state at the same average soilskeleton stress, is a unique function of the bonding effectdue to water menisci at the inter-particle contacts. Thesame result is obtained when examining critical states atdifferent suctions. Based on these observations, an elasto-plastic constitutive model is developed using a single yieldsurface the size of which is controlled by volumetrichardening. In spite of this simplicity, it is shown that themodel reproduces correctly many important features ofunsaturated soil behaviour. It is especially remarkablethat, although only one yield surface is used in theformulation of the model, the irreversible behaviour inwetting–drying cycles is well captured. Because of thebehaviour normalisation achieved by the model, the re-sulting constitutive law is economical in terms of thenumber of tests required for parameter determination.

KEYWORDS: clays; constitutive relations; partial saturation;plasticity; suction

Cet expose presente un modele elasto-plastique pour dessols non satures, modele qui prend explicitement encompte les mecanismes par lesquels la succion affecte lecomportement mecanique ainsi que leur dependance vis-a-vis du degre de saturation. Le modele propose estformule en termes de deux variables constitutives directe-ment liees a ces mecanismes de succion ; la contraintemoyenne du squelette, qui inclut la pression de fluidemoyenne sur les pores du sol et une autre varianteconstitutive scalaire supplementaire, �, liee a la magni-tude de l’effet d’adherence exerce par l’eau menisqueaux contacts entre particules. La formulation du modeleen termes de variables etroitement liees a des mecanismesde comportement specifiques conduit a une unificationremarquable des resultats experimentaux des essais effec-tues avec differentes succions. L’analyse des donneesexperimentales de compression isotrope suggere forte-ment que le quotient entre le taux de pore e d’un sol nonsature et le taux de pore es correspondant a l’etat saturepour la meme contrainte moyenne de squelette de sol, estune fonction unique de l’effet d’adherence du aux menis-ques de l’eau aux contacts entre particules. Le memeresultat est obtenu lorsque l’on examine les etats cri-tiques a differentes succions. En nous basant sur cesobservations, nous avons developpe un modele constitutifelasto-plastique utilisant une unique surface d’ecoulementdont la dimension est commandee par le durcissementvolumetrique. Nous montrons que le modele, malgre sasimplicite, reproduit correctement bien des caracteris-tiques importantes du comportement d’un sol non sature.Il est particulierement frappant de constater que, bienqu’une seule surface d’ecoulement ait ete utilisee dans laformulation du modele, le comportement irreversibledans les cycles mouillage-sechage est bien represente. Enraison de la normalisation de comportement obtenuegrace au modele, la loi constitutive qui en resulte abesoin de moins d’essais pour la determination des para-metres.

INTRODUCTIONSuction has long been recognised as a fundamental vari-

able in the understanding of the mechanical behaviour ofunsaturated soils. For this reasons many well-known consti-tutive models (Alonso et al., 1990; Wheeler & Sivakumar,1995; Cui & Delage, 1996) include suction as a basic stressvariable together with the net stress, � (defined as totalstress minus pore air pressure).

In fact suction influences the mechanical behaviour of anunsaturated soil in two different ways (Karube & Kato,1994; Wheeler & Karube, 1995):

(a) by modifying the skeleton stress through changes in theaverage fluid pressure acting in the soil pores

(b) by providing an additional bonding force at the particlecontacts, often attributed to capillary phenomenaoccurring in the water menisci.

It is important to realise that, for the two mechanisms, theeffects of suction are influenced by the state of saturation ofthe soil. The relative area over which the water and airpressures act depends directly on the degree of saturation(the percentage of pore voids occupied by water), but thesame parameter also affects the number and intensity ofcapillary-induced inter-particle forces.

Therefore models using only suction in their formulationare unlikely to be complete. It is necessary to incorporate,through a parameter such as degree of saturation, informa-tion regarding the proportion of the soil over which suctioneffects are relevant. Therefore it is not surprising thatconstitutive models that use only suction as the unsaturated

Manuscript received 1 May 2002; revised manuscript accepted 20August 2002.Discussion on this paper closes 1 August 2003, for further detailssee p. ii.� Departamento de Ingenierıa del Terreno, Universitat Politecnicade Catalunya, Barcelona, Spain.{ University of Bradford, UK (formerly University of Oxford, UK).

variable face difficulties in describing important features ofunsaturated soil behaviour. Another class of elasto-plasticmodels for unsaturated soils, such as those proposed byJommi & Di Prisco (1994), Bolzon et al. (1996), Loret &Khalili (2000) and Karube & Kawai (2001), are expressed interms of a different set of constitutive variables that expli-citly include the degree of saturation in their definitions. Inthese models the stress variable has the form of the Bishop(1959) stress:

� 9hk ¼ �hk � �hk[ua � �(ua � uw)] (1)

where � 9hk is the Bishop (1959) stress, �hk is the total stress,ua is the air pressure, uw is the water pressure, �hk isKronecker’s delta, and � is a soil parameter that rangesbetween 1 (at saturation) and zero (at dry conditions), andwhich is a function of degree of saturation. The additionalscalar variable is given either by suction or by degree ofsaturation depending on the specific model, with the excep-tion of the model of Karube & Kawai (2001), in which it isa function of both suction and degree of saturation.Although such models explicitly introduce the degree ofsaturation in the definition of the soil variables, they stillpresent limitations when predicting important aspects ofunsaturated soil behaviour unless the additional complexityof multiple yield surfaces is introduced. Two examples offeatures of behaviour that require adequate modelling are:

(a) the irreversible reduction of specific volume that canoccur during drying of an unsaturated soil (that is,during increase of suction)

(b) the dependence of the soil response during virginloading at constant suction on the past history ofsuction variation.

The first type of behaviour has been observed by Alonso etal. (1995) and Sharma (1998) in laboratory tests involvingwetting–drying cycles (that is, cycles of decrease–increaseof suction) on soil samples subjected to oedometric andisotropic conditions respectively. Examples of this type ofsoil response are shown in Fig. 1. The second type ofbehaviour has been observed by Sharma (1998) duringisotropic loading to virgin states of samples at constantsuction. In particular, the results by Sharma (1998) suggestthat a compacted unsaturated soil shows a different stiffnessduring virgin loading at the same constant suction dependingon whether the sample undergoes a wetting–drying cycleprior to loading or not. Fig. 2 shows examples of isotropictests where the dependence of the soil stiffness during virginloading on the previous history of suction variation can beobserved. Moreover, for some of these models the determi-nation of model parameters requires either non-conventionallaboratory tests (for example, tests at constant degree ofsaturation) or a back-analysis for fitting model predictions toconventional laboratory results.

In this paper a model is described that incorporatesexplicitly in its formulation the two distinct suction effectsmentioned previously, including their dependence on degreeof saturation. By staying close to these basic behaviourmechanisms, the proposed elasto-plastic model is capable ofreproducing the most important patterns of unsaturated soilmechanical behaviour, including those indicated above, in arather simple manner employing only a single yield surface.The model also provides an effective way of unifying experi-mental results of tests performed at different suctions. Apartfrom the conceptual benefits of such unification, this factresults in economical procedures for parameter determinationfrom the point of view of the number of laboratory testsrequired.

MODELLING ASSUMPTIONSIn the proposed model the basic stress variable is the

average skeleton stress (Jommi, 2000), which is equivalentto the Bishop (1959) stress where the parameter � of equa-tion (1) is equal to the degree of saturation, Sr:

� 9hk ¼ �hk � �hk[ua � �(ua � uw)] (2)

This variable expresses the average stress acting in the soilskeleton: that is, the difference between the total stress andthe average pressure of the two fluid phases (i.e. gas andliquid), with the degree of saturation as a weighting para-meter. It therefore incorporates in a direct manner the firstof the suction roles noted above. The definition of theaverage skeleton stress represents a natural extension to theunsaturated domain of the Terzaghi (1936) effective stressfor saturated granular materials, and it reduces to theTerzaghi effective stress at saturated condition (that is,degree of saturation equal to unity).

0

�4

�8

�12

Vol

umet

ric s

trai

n: %

First wetting

First wetting

Second wetting

Third wettingSecond drying

3 10 100 1000 10000 100000

Suction: kPa

(b)

1.4

1.2

1

0.8

Voi

d ra

tio

3 10 100 1000

Suction: kPa

Second drying

Second wetting

First wetting

First drying

(a)

Fig. 1. Wetting–drying cycle on: (a) compacted mixture ofbentonite and kaolin under constant isotropic net stress of10 kPa (after Sharma, 1998); (b) compacted Boom clay underoedometric conditions at constant vertical stress of 400 kPa(after Alonso et al., 1995)

124 GALLIPOLI, GENS, SHARMA AND VAUNAT

Laboratory tests have shown, however, that it is notpossible to explain important features of the behaviour ofunsaturated soils, such as the irreversible compression (col-lapse) during wetting (that is, during a suction reduction)and the increase of the pre-consolidation pressure with in-creasing suction, by using the average skeleton stress as theonly constitutive variable (e.g. Jennings & Burland, 1962).To account for these phenomena it is necessary to considerthe second suction mechanism. The irreversible mechanicalresponse of a granular material is associated mainly with therelative slippage taking place at the interface between soilparticles. In an unsaturated soil the possibility of suchslippage is partially reduced by the stabilising effect of thenormal force exerted at the inter-particle contacts by menis-cus lenses of water at negative pressure (Wheeler & Karube,1995). Several features of the elasto-plastic behaviour ofunsaturated soil are therefore likely to be the consequence of

bonding and de-bonding phenomena between soil particlesdue to the formation and vanishing of water menisci atinter-particle contacts, and they cannot be accounted for byusing exclusively the average skeleton stress as a constitutivevariable.

Consequently an additional constitutive variable, �, needsto be introduced as a measure of the magnitude of the inter-particle bonding due to water menisci so that the secondtype of suction effect is properly accounted for. The magni-tude of such inter-particle bonding is expected to be theresult of two contributions:

(a) the number of water menisci per unit volume of thesolid fraction

(b) the intensity of the stabilising normal force exerted atthe inter-particle contact by a single water meniscus.

Hence the variable � is defined in the present formulation asthe product of two factors: the degree of saturation of theair, (1� Sr), and the function of suction, f (s):

� ¼ f (s)(1� Sr) (3)

The factor (1� Sr) accounts for the number of watermenisci per unit volume of solid fraction. The existence of aunique relationship between the value of (1� Sr) and thenumber of water menisci per unit volume of solid fraction isa physically reasonable assumption; however, the uniquenessof such relationship is rigorously true only for the ideal casewhere the soil is rigid (that is, when the dimensions andshapes of voids do not change as a result particle rearrange-ments), and where each value of degree of saturationcorresponds to a given arrangement of water within soilpores. The term (1� Sr) is equal to zero when the soil issaturated (that is, Sr ¼ 1) and water menisci are absent,whereas it assumes positive increasing values when the num-ber of water menisci increases. The number of water menisciper unit volume of solid fraction can therefore be expressedas a monotonic increasing function of the term (1� Sr). Thevalidity of this definition does not apply to the case of a soilin an extremely dry state, when the water menisci will startto disappear from the particle contacts. Although the exten-sion to the case of extremely dry soils should not presentany conceptual difficulty, this is not covered in the presentpaper because the experimental validation would requireexperimental results from a test programme conducted onsamples at very low degree of saturation, which are cur-rently unavailable.

Clearly, the relationship between the number of watermenisci per unit volume of solid fraction and the term(1� Sr) is dependent on the specific fabric of the soil (thatis, on the pore size distribution of the soil). However, for thepurposes of this work it is not necessary to characterise sucha relationship explicitly, because this information is implicitin the definition of the function, introduced later in thepaper, that provides the variation of the ratio e=es in termsof the bonding variable �.

The function of suction f (s), which multiplies the factor(1� Sr), varies monotonically between 1 and 1·5 for valuesof suction ranging between zero and infinity respectively,and it accounts for the increase with increasing suction ofthe stabilising inter-particle force exerted by a single menis-cus. In particular, it expresses the ratio between the value ofstabilising force at a given suction, s, and the value ofstabilising force at a suction of zero for the ideal case of awater meniscus located at the contact between two identicalspheres (the analytical solution of this problem is due toFisher, 1926). The specific form of the function f (s)depends on the size of the spheres and on the value of thewater surface tension, but the range of variation, between 1and 1·5, is always the same regardless of dimensions and

1.3

1.2

1.1

1

0.9

Voi

d ra

tio

Before wetting–drying cycle

After wetting–drying cycle

5 10 20 30 50 100 200 300 500

Mean net stress: kPa(a)

1.2

1.1

1

0.9

Voi

d ra

tio

Before wetting–drying cycle

After wetting–drying cycle

5 10 20 30 50 100 200 300 500

Mean net stress: kPa(b)

Fig. 2. Isotropic virgin loading of compacted mixture ofbentonite and kaolin at: (a) constant suction of 300 kPa (afterSharma, 1998); (b) constant suction of 200 kPa (after Sharma,1998)

ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 125

physical properties. The relationship f (s) used in this workis shown in Fig. 3, corresponding to the case of two sphereshaving radii of 1 �m and a value of the surface tension ofwater corresponding to a temperature of 208C. Haines(1925) suggested that a material with the texture of acompacted kaolin could be represented by spheres havingradii equal to 1 �m. Obviously the shapes of the aggregatesare far from being spheres of the same size. In addition, forsoils with a multi-modal pore size distribution, the dimen-sion of the spherical grains in the solution of Fisher (1926)should be defined as a variable depending on the averagesize of the soil pores that include water menisci. At thisstage of development of the model, however, the assumptionof a simplified relationship, such as the one given in Fig. 3,is considered reasonable.

The presence of meniscus water provides a physicalexplanation for the experimental observation that, at thesame value of average skeleton stress, the value of void ratioduring virgin loading of unsaturated soil is always greaterthan the value for the same soil subjected to the same loadunder saturated conditions. The existence of water in theform of meniscus lenses within an unsaturated soil makesthe inter-particle contacts more stable, and therefore restrainsthe reciprocal slippage of soil particles that causes compres-sive strains during virgin loading. Consistent with suchempirical observations, this work introduces a fundamentalmodelling assumption specifying that, during virgin loadingof an unsaturated soil, the ratio e=es between void ratio inunsaturated conditions, e, and void ratio in saturated condi-tions, es, at the same average skeleton stress state is aunique function of the bonding variable, �. This assumptionnot only provides an essential starting point for the develop-ment of the model, it also offers a powerful unifying per-spective to examine the results of tests performed atdifferent suctions. This assumption is validated in the nextsection on the basis of published laboratory test data.

EXPERIMENTAL VALIDATION OF MODELLINGASSUMPTIONS

The validation of the assumption introduced in the pre-vious section has involved the analysis of different sets of

data from laboratory tests performed on compacted Spes-white kaolin (Sivakumar, 1993; Wheeler & Sivakumar,2000), on a compacted mixture of bentonite and kaolin(Sharma, 1998), and on compacted Kiunyu gravel (Toll,1990). The first part of this section analyses the data fromisotropic virgin compression tests at constant suction(Sivakumar, 1993; Sharma, 1998). At the end of the section,the analysis of further experimental data from triaxial sheartests on compacted Speswhite kaolin (Sivakumar, 1993;Wheeler & Sivakumar, 2000) and on compacted Kiunyugravel (Toll, 1990) demonstrates that the conclusionachieved for isotropic stress states can also be extended tonon-isotropic stress states.

Sivakumar (1993) and Sharma (1998) performed isotropicvirgin compression of soil samples at different values ofsuction—100 kPa, 200 kPa and 300 kPa—as well as of satu-rated samples. During these tests the corresponding changesof void ratio, e, and water ratio, ew (that is, the volume ofwater in a volume of soil containing unit volume of solids),were measured. The analysis of the experimental resultsindicates that, for the range of stresses considered, thenormal compression lines at constant suction follow a linearrelationship in the semi-logarithmic planes e� ln p andew � ln p (where p is the isotropic net stress). Each normalcompression line is therefore identified by the values of thetwo parameters that correspond to the slope and to theintercept at a given value of p. As for the data set bySharma (1998) there were no virgin loading tests on satu-rated samples; the slope and intercept of the saturatednormal compression line were estimated from the dryingbranch of a wetting–drying test under isotropic constantload. In this type of test, after an initial wetting that broughtthe soil to saturation, the sample was subjected to dryingthat caused significant irreversible changes of void ratio. Theprinciple of effective stress holds during most of such dryingbecause the sample remained saturated for a large increaseof suction owing to its high air-entry value. Under saturatedconditions the imposed change of suction corresponds to anequivalent change of the effective stress. Hence, by plottingthe void ratio against the isotropic effective stress, it waspossible to estimate the slope and the intercept of thesaturated normal compression line.

The values of slopes and intercepts of normal compressionlines of e and ew at constant suction were used to re-plotthe normal compression lines in terms of the isotropicaverage skeleton stress, p 0. Figs 4 and 5 show the normalcompression lines at constant suctions of zero (saturated),100 kPa, 200 kPa and 300 kPa, in the semi-logarithmic planee� ln p 0 for each set of data respectively.

Inspection of Figs 4 and 5 reveals that the normal com-pression lines at non-zero values of suction are not straightlines in the semi-logarithmic plane e� ln p 0, but they arecurves with slopes that decrease as they approach thesaturated line (zero suction). This is consistent with theexperimental observation that the degree of saturation in-creases during isotropic loading to virgin states at constantsuction. Indeed, if a soil sample attains saturation duringcompression at a positive value of suction, the isotropicaverage skeleton stress coincides with the saturated effectivestress, and the corresponding value of void ratio should lieon the saturated normal compression line. After saturation,the normal compression line at non-zero suction shouldtherefore have the same slope as the saturated normal com-pression line. It is therefore to be expected that the slope ofthe normal compression lines at non-zero suction progres-sively reduces as they converge towards the saturated line.

From the normal compression lines in the semi-logarith-mic plane e� ln p 0 shown in Figs 4 and 5 it is possible tocalculate the ratio between the e value of the unsaturated

1.5

1.4

1.3

1.2

1.1

1

f(s)

0 1000 2000 3000 4000

s: kPa

Fig. 3. Ratio between inter-particle forces at suction s and atnull suction due to a water meniscus located at the contactbetween two identical spheres (analytical solution by Fisher,1926)

126 GALLIPOLI, GENS, SHARMA AND VAUNAT

soil and that corresponding to the saturated state, es, at thesame average skeleton stress. Figs 6 and 7 show, for the datasets of Sivakumar (1993) and Sharma (1998) respectively,the value of the ratio e=es plotted against the value of thebonding variable, �, defined by equation (3) (correspondingto the values of Sr and f (s) of the unsaturated soil). Thevalue of the function of suction, f (s), has been calculatedaccording to the relationship shown in Fig. 3, and it is equalto 1·10, 1·15 and 1·18 for suction values of 100 kPa,200 kPa and 300 kPa respectively. The relationship shownin Fig. 3 refers to the Fisher (1926) solution where thespherical grains have radii equal to 1 �m. This is the orderof magnitude of the macrostructural voids of a clay soilcompacted dry of optimum, such as the soils investigated bySivakumar (1993) and Sharma (1998). Porosimetry studieshave shown that clay materials compacted dry of optimum

present a marked bimodal pore size distribution (e.g. Gens& Alonso, 1992) with macrostructural and microstructuralvoids of the order of magnitude of 1 �m and 0·01 �mrespectively. For the range of suctions investigated bySivakumar (1993) and Sharma (1998) it is reasonable toexpect that only macro-voids are affected by desaturation(and hence by the formation of water menisci) while themicro-voids stay saturated.

Inspection of Figs 6 and 7 suggests remarkably that, forall the three values of suction investigated, the data fromnormal compression are consistent with a unique relationshiplinking the value of the proportion e=es and the bondingvariable, �. Such a bonding variable therefore appears to beuniquely related to the ability of the skeleton to sustain

1.4

1.3

1.2

1.1

1

0.9

0.8

e

s � 0 kPa (saturated)

s � 100 kPa

s � 200 kPa

s � 300 kPa

40 50 60 70 80 90100 200 300 400 500p″: kPa

Fig. 4. Normal compression lines at constant suction in theplane e� ln p 0 (data by Sharma, 1998)

1.4

1.3

1.2

1.1

1

0.9

0.8

e

s � 0 kPa (saturated)

s � 100 kPa

s � 200 kPa

s � 300 kPa

40 50 60 70 80 90100 200 300 400 500p″: kPa

Fig. 5. Normal compression lines at constant suction in theplane e� ln p 0 (data by Sivakumar, 1993)

Model equationExperimental (s � 100 kPa)Experimental (s � 200 kPa)

Experimental (s � 300 kPa)

1.4

1.3

1.2

1.1

1

e/e s

0.60 0.2 0.4�

Fig. 6. Relationship between ratio e=es and bonding factor �during isotropic virgin loading at constant suction (data bySharma, 1998)

Model equationExperimental (s � 100 kPa)

Experimental (s � 200 kPa)Experimental (s � 300 kPa)

1.4

1.3

1.2

1.1

1

e/e s

0.60 0.2 0.4�

Fig. 7. Relationship between ratio e=es and bonding factor �during isotropic virgin loading at constant suction (data bySivakumar, 1993)

ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 127

higher void ratios when the soil is under suction. For eachof the three curves at constant suction shown in Figs 6 and7, the value of the proportion e=es is expected to attain avalue of 1 when � is equal to zero (that is, when the sampleachieves saturation) because in this case the normal com-pression lines at non-zero suctions coincide with the satu-rated line in the semi-logarithmic plane e� ln p 0. Themodel equation that fits the three curves of e=es against � atconstant suction in Figs 6 and 7 has the following form:

e

es

¼ 1� a � [1� exp (b � �)] (4)

where a and b are fitting parameters. Equation (4) predicts avalue of e=es equal to 1 when � is equal to zero, consistentwith the physical explanation given above.

For the range of suction investigated, the value of thefunction f (s) varies relatively little in comparison with thevariation of the value of the bonding variable �. It is there-fore reasonable to expect that experimental data might besimilarly consistent with a relationship linking the value ofthe proportion e=es during isotropic virgin loading to thevalue of the degree of saturation of the gas phase, (1� Sr).

Sivakumar (1993) and Wheeler & Sivakumar (2000) pre-sented further experimental data from shearing tests tocritical state on compacted Speswhite kaolin under varioussuction values. These data have been used to investigatewhether the relationship between the ratio e=es and thebonding variable � could be extended to non-isotropic stressstates. The model equation in Fig. 7, which had been definedon the basis of isotropic normal compression tests, wastherefore used to predict values of void ratio at criticalstates. The predicted values of void ratio at critical statewere computed following the same procedure as in theisotropic case. First the saturated critical-state line in thesemi-logarithmic plane (e� ln p 0) was defined (by fixing itsslope and intercept) on the basis of shearing tests performedby Sivakumar (1993) and Wheeler & Sivakumar (2000) onsaturated samples. Then, for each unsaturated samplesheared to critical state, the corresponding experimentalvalues of isotropic net stress, degree of saturation andsuction at critical state were used to calculate the isotropicaverage skeleton stress, p 0, and the bonding variable �.These values of p 0 and � were then employed to computethe void ratio, es, from the saturated critical-state lineand the ratio e=es from the model equation of Fig. 7respectively. Fig. 8 shows the comparison between predictedand experimental values of void ratio at critical statecorresponding to different suction levels.

Inspection of Fig. 8 indicates remarkably that the relation-ship established between the ratio e=es and the bondingfactor � for isotropic virgin compression and given by equa-tion (4) can also accurately predict the void ratio values atcritical state. This implies that such a relationship might beunique for the elasto-plastic loading of an unsaturated soilregardless of the specific stress ratio applied to the sample,and that the selected bonding variable closely represents thereal effect of suction on inter-granular stress. The differentseries of Fig. 8 correspond to different procedures adoptedby Wheeler & Sivakumar (2000) for the compaction ofSpeswhite kaolin at the same dry of optimum water content(that is, for the test series II and III the compaction pressureand method of compaction were different from those em-ployed for preparing the samples from series I shown in Figs5 and 7). On the basis of their empirical results, Wheeler &Sivakumar (2000) concluded that the behaviour at criticalstate of a soil compacted at the same dry of optimum watercontent is not affected by the procedure adopted for compac-tion. This result is also confirmed by the comparison shownin Fig. 8.

Finally, the analysis of the data from undrained (with respectto the water phase) triaxial shear tests performed by Toll(1990) on compacted samples of a lateritic gravel fromKenya (Kiunyu gravel) is presented. Samples were com-pacted by Toll (1990) at different values of water contentranging from 17·0 to 27·7 and then sheared in axial com-pression to critical state while preventing the flow of water.The author reports the measured values of the void ratio,degree of saturation, suction and net stress state at the endof the tests when the unsaturated samples have attainedcritical state conditions. Together with the results fromunsaturated soil samples, the author presents a smaller set ofdata from undrained triaxial shear tests to critical stateperformed on saturated samples. For the saturated tests, thesoil samples were compacted at water content ranging be-tween 18·7 and 31·0 and were then saturated prior to testing.On the basis of the saturated shear tests, Toll (1990) sug-gests the values of the slope and intercept of saturatedcritical-state line in the semi-logarithmic plane (e� ln p 0).

The data presented by Toll (1990) are limited to thecritical-state values measured at the end of the undrainedshearing of each sample. Such data were used in this workto validate the proposed assumption of a unique relationshipbetween the ratio e=es (corresponding to a given value ofthe isotropic average skeleton stress) and the bonding vari-able � at the critical state. The soil tested by Toll (1990) islikely to exhibit a grading different from that of the soilsinvestigated by Sivakumar (1993) and Sharma (1998). How-ever, owing to the absence of precise information on thefabric of the soil tested by Toll (1990), the radii of thespheres in the Fisher (1926) solution (that is, in the functionf (s) of equation (3)) were taken equal to 1 �m, the samevalue as in the previous two analyses. For each experimentaldata point the value of the function f (s) was then calculatedaccording to the suction measured by Toll (1990) at thecritical state.

For the purposes of the study presented here, the unsatu-rated samples tested by Toll (1990) were classified in twodifferent groups, each one including samples compacted atsimilar values of water content. The data shown in Fig. 9refer to unsaturated samples whose compaction water con-

0.8

0.9

1

1.1

1.2

Pre

dict

ed e

Identity functions � 100 kPa (series I)s � 200 kPa (series I)s � 300 kPa (series I)s � 100 kPa (series II)s � 300 kPa (series II)s � 100 kPa (series III)s � 300 kPa (series III)

0.8 0.9 1 1.1 1.2

Experimental e

Fig. 8. Comparison between experimental and predicted voidratio at critical state (data by Sivakumar, 1993 and Wheeler &Sivakumar, 2000)

128 GALLIPOLI, GENS, SHARMA AND VAUNAT

tent ranged between 24·9% and 27·7%, whereas for thesamples shown in Fig. 10 the compaction water contentranged between 19·6% and 21·9%. The value of the compac-tion water content significantly affects the ensuing fabric ofthe unsaturated sample, and this in turn has an effect on themechanical response of the sample, including the responseat critical state (as discussed by Wheeler & Sivakumar,2000). Hence the purpose of such a distinction was to selecttwo homogeneous groups of samples that presented a similarsoil fabric, and whose mechanical response during laboratorytesting could therefore be compared. From examination ofFigs 9 and 10 it can be observed that all critical-state datapoints (each of them corresponding to different values of

suction, net stress state and degree of saturation at criticalstate) follow a unique trend when plotted in the plane(e=es, �). In Figs 9 and 10 the expression of the curveinterpolating the experimental data is given by equation (4)(the same expression as used for the experimental dataof Sharma (1998) and Sivakumar (1993) shown in Figs 6and 7).

The samples compacted at water contents between 24·9%and 27·7% (Fig. 9) had suction values ranging from 2 kPa to73 kPa on reaching the critical state, whereas the suctionvalues at critical state for the samples compacted at watercontents between 19·6% and 21·9% (Fig. 10) varied between22 kPa and 537 kPa. The proposed relationship, therefore, isshown to be capable of bringing together results from a verywide range of suction values.

FORMULATION OF THE ELASTO-PLASTIC STRESS–STRAIN MODEL

An elasto-plastic isotropic stress–strain model for unsatu-rated soils incorporating volumetric hardening is describedin this section. The success of the modelling ideas proposedhere will be demonstrated in the next section by comparingthe predictions with the experimental results from varioustypes of laboratory test, all performed under isotropic load-ing. The development of the model is hence limited in thissection to isotropic stress states in order to concentrateattention on the basic features of the model. Extension tomore general stress states is quite straightforward followingstandard procedures (Gens, 1995).

The formulation of a constitutive model including volu-metric hardening requires the definition of:

(a) a normal compression state surface, which relates thevalues of void ratio, e, isotropic average skeleton stress,p 0, and bonding variable, �, during the irreversiblebehaviour of the soil

(b) an incremental expression that relates the elastic part ofthe change of void ratio, e, to the changes of theisotropic average skeleton stress, p 0, and bondingvariable, �.

The normal compression state surface is defined here as theproduct of two factors. The first factor is the equation of thesaturated normal compression line relating the variation ofthe void ratio, es, to the change of the isotropic averageskeleton stress, p 0, and the second factor is the equation thatlinks the variation of the ratio e=es to the change of thebonding variable, �. For the materials studied here, theanalytical form of the normal compression state surface istherefore expressed as

e( p 0, �) ¼ e

es

(�)es( p 0) (5)

where e( p 0, �) is the normal compression state surface,(e=es)(�) is given by equation (4), and es( p 0) is the satur-ated normal compression line (a straight line in the semi-logarithmic plane e� ln p 0) having the form

es( p 0) ¼ N � º ln p 0 (6)

N and º in equation (6) are the intercept (at p 0 ¼ 1 kPa)and the slope of the saturated normal compression linerespectively. Note that, for saturated conditions, the isotropicaverage skeleton stress, p 0, coincides with the isotropiceffective stress, p, and therefore the parameters N and º areequal to those that identify the saturated normal compressionline in the semi-logarithmic plane e� ln p. Fig. 11(a) showsthree examples of normal compression lines that lie on thenormal compression state surface and correspond to constantvalues of the bonding variable, �. Equations (5) and (6)

1.4

1.3

1.2

1.1

1

0.950 0.1 0.2 0.3 0.4

�

e/e s

Experimental

Model equation

Fig. 9. Relationship between ratio e=es and bonding factor � atcritical state for soil samples compacted at water contentbetween 24·9% and 27·7%. The suctions at critical state rangefrom 2 kPa to 73 kPa (data by Toll, 1990)

1.3

1.2

1.1

1

0.950 0.2 0.60.4

�

e/e s

Experimental

Model equation

1.4

1.5

Fig. 10. Relationship between ratio e=es and bonding factor � atcritical state for soil samples compacted at water contentbetween 19·6% and 21·9%. The suctions at critical state rangefrom 22 kPa to 537 kPa (data by Toll, 1990)

ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 129

indicate that the normal compression lines at constant � arestraight lines in the semi-logarithmic plane e� ln p 0 andthat their extrapolations at high values of p 0 intersect eachother at the same point, coinciding with a value of the voidratio es on the saturated normal compression line equal tozero.

The elastic change of void ratio, ˜ee, is assumed to begiven by

˜ee ¼ �k lnp 0f

p9i(7)

where k is the elastic swelling index and p 0i to p 0f are theinitial and final value of the isotropic average soil skeletonstress respectively. Equation (7) implies that the elasticchange of void ratio depends exclusively on the change ofthe isotropic average skeleton stress, p 0 (that is, it is

independent of the variation of the bonding variable �). Thisis equivalent to assuming that the elastic deformation of thesoil skeleton is not affected by the bonding action that thewater menisci exert at the inter-particle contacts. It will beshown in the next section that this assumption fits well theelastic behaviour of the laboratory tests considered in thiswork.

The normal compression state surface defined by equation(5) acts as a limiting surface in (e, p 0, �) space, where itseparates the region of attainable soil states from the regionof non-attainable soil states. The soil response is elasticwhile the soil follows a path inside the space of attainablesoil states. When the soil path reaches the normal compres-sion state surface, this surface imposes a constraint onfurther changes of e, p 0 and �, and the soil state cantherefore either move back inside the space of the attainablesoil states or follow a path lying on the normal compressionstate surface. When the latter possibility occurs, irreversible(elasto-plastic) changes of void ratio develop.

Thus the normal compression state surface and the elasticlaw introduced above (equations (5) and (7) respectively)implicitly define a yield locus that incorporates a volumetrichardening rule. To obtain the analytical form of such a yieldlocus consider the elastic stress path in Fig. 11(a) startingfrom the soil state denoted by 1, at an isotropic averageskeleton stress p 0o(0) on the saturated normal compressionline, and moving to the soil state denoted by 2, at anisotropic average skeleton stress p 0o(�2) on the unsaturatednormal compression line corresponding to � ¼ �2. Thechange of void ratio during the path from state 1 to state 2is computed according to the elastic equation (7):

˜e ¼ �k lnp 0o(�1)

p 0o(0)(8)

As the soil states 1 and 2 also belong to the normalcompression state surface they must lie on the same yieldlocus, and an alternative expression for the variation of voidratio during the path from state 1 to state 2 can therefore beobtained by using the normal consolidation state surface ofequation (5):

˜e ¼ e[ p 0(0), 0]� e[ p 0(�1), �1] ¼ N � º ln p 0o(0)

� e

es

(�1)[N � º ln p 0o(�1)] (9)

By equating equation (8) and equation (9) and then rearran-ging, the following equation of the yield locus in theisotropic plane �� ln p 0 is obtained:

ln p 0o(�1) ¼ º� ke

es

(�)º� kln p 0o(0)þ

e

es

(�)� 1

� �(1þ N )

e

es

(�)º� k

(10)

Figure 11(b) shows the yield locus of equation (10) in theisotropic plane �� ln p 0 together with the two yield pointscorresponding to the soil states 1 and 2, which are identifiedby the coordinates ( p 0o(0), 0) and ( p 0o(�2), �2) respectively.

Figure 11(b) also shows an expanded yield locus, indi-cated by the broken line, which refers to a soil sample thathas experienced additional plastic volumetric strains andwhose yield locus has therefore undergone volumetric hard-ening. The current size of the yield locus is identified by thevalue of its intercept p 0o(0) with the horizontal axis, whichis the yield value of the isotropic average skeleton stressduring isotropic compression of a saturated sample. Thesaturated yield stress p 0o(0) can therefore be assumed as the

e�2 � �1

�1 � 0

� � 0

Normal compression lines

Elastic stress path

1

2

p″0(0)p″

p″0(�2)

(a)

� � 0

� � �2

�

Initial yield locus

Expanded yield locus

2

1

p″0(0)p″

p″0(�2)

(b)

Fig. 11. Derivation of the yield locus in the isotropic plane: (a)change of void ratio; (b) stress path

130 GALLIPOLI, GENS, SHARMA AND VAUNAT

hardening parameter of the present elasto-plastic model.The irreversible change of void ratio, ˜ep, associated withthe expansion of the yield locus from an initial positionidentified by p 0o(0) ¼ p 0o(0)i to a final position identified byp 0o(0) ¼ p 0o(0)f coincides then with the irreversible changeof void ratio calculated by the saturated normal compressionline for a variation of the isotropic average skeleton stressfrom p 0o(0)i to p 0o(0)f :

˜ep ¼ �(º� k) lnp 0o(0)f

p 0o(0)i

(11)

and equation (11) thus represents the volumetric hardeningrule of the proposed elasto-plastic model.

The complete model includes a formulation to computethe degree of saturation that must incorporate the effect ofhydraulic hysteresis and stress-induced changes of soil fab-ric. The relationships proposed by Vaunat et al. (2000) andGallipoli et al. (2003) can be used for this purpose, but adetailed description of this component of the model isoutside the scope of the paper. For the model computationspresented in the next section, the experimentally observeddegrees of saturation have been used. In this way thedifferences between predictions and observations must beattributed exclusively to the mechanical elasto-plastic model.

MODEL PREDICTIONSThe good performance of the proposed elasto-plastic

model is demonstrated here by comparing the results from aselection of experiments performed by Sharma (1998) on acompacted mixture of bentonite and kaolin with the corre-sponding model predictions. In particular, the comparisonwill show the potential of the proposed model for correctlypredicting:

(a) The initial yield locus of the soil corresponding to theafter-compaction state

(b) The irreversible change of void ratio occurring duringwetting (collapse)

(c) The irreversible change of void ratio during drying(d ) the dependence of the soil response during isotropic

virgin loading at constant suction on the previoushistory of suction variation.

Points (c) and (d ) refer to typical features of unsaturated soilbehaviour that are not taken into account by existing elasto-plastic constitutive frameworks formulated in terms of asingle yield surface.

The selection process of the model parameter values usedfor the predictions (see Table 1) has been described earlier,except for the value of the elastic swelling index, k, whichwas selected on the basis of elastic isotropic loading–unloading cycles at constant suction.

Figures 12–14 show the comparison between experimentaland predicted behaviour for three isotropic loading tests atconstant suction (100 kPa, 200 kPa and 300 kPa respectively)that involve elasto-plastic yielding. Inspection of Figs 12–14reveals that the proposed model correctly calculates the

respective yield points by assuming for all three test simula-tions the same initial yield locus associated with a value ofthe hardening parameter, p 0o(0) ¼ 17 kPa. Such a modelprediction is corroborated by the soil response observed bySharma (1998) during the equalisation stage prior to loading,when the suction of the three samples was decreased fromthe value after compaction to 100 kPa, 200 kPa and 300 kParespectively. During this stage all three samples experiencedexclusively elastic swelling, which indicates that the initialyield curve after compaction had not undergone furtherexpansion associated with plastic volumetric compression(collapse). It is then expected that all three samples wouldyield on the same locus during isotropic loading, and theproposed model indeed correctly predicts this. Therefore, forthe test simulations presented in the remainder of thissection, the value of the hardening parameter correspondingto the soil after compaction was assumed to be equal to17 kPa.

Table 1. Parameter values for the proposed elasto-plastic model

Parameter Value

º, slope of NCL at s ¼ 0 0·144N , e on NCL at s ¼ 0 for p 0 ¼ 1 kPa 1·759k, swelling index for changes of p 0 0·040a, parameter of model equation (4) 0·369b, parameter of model equation (4) 1·419

1.4

1.3

1.2

1.1

1

0.9

e

Experimental

Prediction

5 10 20 30 50 100 200 300 500p: kPa

(a)

Initial yield locus

Expanded yield locus

0.5

0.4

0.3

0.2

0.1

0

�

0 10 100 1000p″: kPa

(b)

Fig. 12. Model prediction for isotropic virgin loading atconstant suction of 100 kPa (experimental data by Sharma,1998): (a) change of void ratio; (b) stress path

ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 131

Figure 13 also shows the comparison between modelpredictions and experimental results for an elastic loading–unloading cycle, which confirms the adequacy of the elasticlaw given by equation (7). The very good match betweenexperimental and predicted values of void ratio at thebeginning of the loading in Figs 12–14 provides a furtherproof of the validity of equation (7) because the predictedinitial values of void ratio are computed by means of anelastic path starting from the saturated yield stress state( p 0o(0) ¼ 17 kPa) according to the following expression:

ei ¼ eo � k lnp 0i

17 kPa(12)

where eo is the void ratio predicted by the saturated normalcompression line for p 0 ¼ 17 kPa and p 0i is the isotropicaverage skeleton stress of the unsaturated sample at thebeginning of loading.

Figure 15 shows the comparison between experimentaland predicted behaviour for a wetting–drying cycle per-

formed at a constant isotropic net stress of 50 kPa. Inspec-tion of the stress path followed by the soil during the test inFig. 15(b) reveals that the model predicts irreversiblechanges of void ratio during both the wetting and the dryingbranch of the test. Yielding of the soil occurs initially duringwetting, and the consequent development of elasto-plasticstrains produces the first expansion of the yield locus fromits initial position to the position indicated by (A) (corre-sponding to the end of wetting). After the reversal of suctionthe model continues to predict elasto-plastic deformationsduring the whole drying, and this corresponds to a furtherexpansion of the yield locus from position (A) to the finalposition (B). This test simulation clearly demonstrates thepotential of the present framework to interpret the elasto-plastic volumetric strains that occur during both the wettingand the drying phases as a single mechanical phenomenonthat can be modelled by employing only one yield locus.Part of the discrepancy between experimental results andmodel prediction in Fig. 15(a) is due to the incomplete

1.4

1.3

1.2

1.1

1

0.9

0.5

0.4

0.3

0.2

0.1

0

�e

First loading

Unloading–reloading

(A)

Second loading

(B)

Experimental

Prediction

5 10 20 30 50 100 200 300 500

p: kPa(a)

Initial yield locus

Expanded yield locus

First loading

Unloading–reloading

(A)

(B)

Second loading

5 10 100 1000

p″: kPa(b)

Fig. 13. Model prediction for isotropic virgin loading atconstant suction of 200 kPa (experimental data by Sharma,1998): (a) change of void ratio; (b) stress path

1.4

1.3

1.2

1.1

1

0.9

e

Experimental

Prediction

5 10 20 30 50 100 200 300 500p: kPa

(a)

Initial yield locus

Expanded yield locus

0.5

0.4

0.3

0.2

0.1

0

�

5 10 100 1000

p″: kPa(b)

Fig. 14. Model prediction for isotropic virgin loading atconstant suction of 300 kPa (experimental data by Sharma,1998): (a) change of void ratio; (b) stress path

132 GALLIPOLI, GENS, SHARMA AND VAUNAT

equalisation of suction within the sample during the test.The occurrence of incomplete equalisation is proven by theexperimental observation, reported by Sharma (1998), that asignificant amount of water flowed into the sample duringthe stabilisation period at the end of the wetting stage, whenthe sample was kept at constant suction of 100 kPa for aperiod of time necessary to equalise suction before subse-quent drying (the vertical change of void ratio at constantsuction of 100 kPa shown in Fig. 15(a) corresponds to thisstabilisation period). However, despite such experimentallimitations, inspection of Fig. 15(a) still indicates a satisfac-tory agreement between predicted and computed results.

Now the more complex stress paths involving wetting–drying cycles are considered. Fig. 16 shows the comparisonbetween the experimental and predicted behaviour duringwetting–drying cycles performed at a constant isotropic netstress of 10 kPa. Inspection of Fig. 16(b) indicates thatelasto-plastic strains occur exclusively during the dryingphases whereas elastic swelling takes place during the

wetting phases. The irreversible strains generated by the firstdrying produce an expansion of the yield locus from theinitial position to position (A) whereas the second dryingoriginates a further expansion from position (A) to position(B). Note that, as explained above, the discontinuity in theslope of the wetting paths shown in Fig. 16(b) is due to thestabilisation phase following incomplete suction equalisationduring previous wetting.

The model predictions in Figs 15 and 16 represent asignificant improvement over existing elasto-plastic modelsbased on a single yield locus, which would incorrectlypredict elastic compression during all the drying phases ofthe above tests.

Finally, Fig. 17 shows the comparison between experimen-tal and predicted behaviour for two constant suction isotro-pic tests that show different mechanical responses dependingon whether or not the sample has undergone a wetting–drying cycle prior to loading. Inspection of Fig. 17 revealsthat the model is capable of capturing the different stiff-nesses shown by the soil during virgin loading in the two

1.4

1.3

1.2

1.1

1

0.9

e

Wetting

(A)

Drying

(B)

Experimental

Prediction

50 100 200 300 500s: kPa

(a)

Initial yield locus

Expanded yield locus

Wetting

Drying

(A)

(B)

0.5

0.4

0.3

0.2

0.1

0.0

�

5 10 100 1000

p″: kPa(b)

Fig. 15. Model prediction for a wetting–drying cycle of asample subjected to a constant isotropic net stress of 50 kPa(experimental data by Sharma, 1998): (a) change of void ratio;(b) stress path

Initial yield locus

Expanded yield locus

First wetting

Second wetting

First drying

Second drying

(B)

(A)

0.6

0.4

0.2

0

�

5 10 100 1000p″: kPa

(b)

1.4

1.2

1

0.8

Experimental

Prediction

First wetting

First drying

Second wetting

(A)

Second drying

(B)

3 10 100 1000s: kPa

(a)

e

Fig. 16. Model prediction for wetting–drying cycles of a samplesubjected to a constant isotropic net stress of 10 kPa (experi-mental data by Sharma, 1998): (a) change of void ratio; (b)stress path

ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 133

cases. This is a significant advance with respect to existingmodels that would instead predict the same slope of thenormal compression lines for both cases regardless of theprevious history of suction variation. A further improvementwith respect to existing models is that the present frameworkcorrectly predicts different values of void ratio at the begin-ning of loading in the two cases of Fig. 15(a). This is due tothe irreversible change of void ratio that occurs duringdrying of the sample subjected to the wetting–drying cycle(see Fig. 15(b)), which is also reflected in the differentdegree of expansion of the yield locus achieved at the endof test in the two cases (position (A) and position (B)respectively).

CONCLUSIONSThe paper proposes an innovative constitutive framework

for unsaturated soil that is able to explain the various mech-

anical features of this material by resorting to a physicaldescription of the different effects of suction on soil strain-ing. In the assumed mechanism, the relative slippage of soilparticles is governed by two counteracting actions exerted onthe assemblage of soil particles:

(a) the perturbing action of the average stress state actingon the soil skeleton

(b) the stabilising action of the normal force exerted at theinter-particle contacts by water menisci.

The variables controlling each one of these actions (that is,the average skeleton stress variable, � 0, and the bondingvariable, �, respectively) are defined on the basis of thecurrent values of the net stress state, suction and degree ofsaturation. The introduction of degree of saturation in thedefinition of the soil constitutive variables is essential torepresent properly the contribution of soil suction to the twoeffects described above.

Based on a physical argument, the present proposalassumes that, during the elasto-plastic loading of a soilelement, the proportion e=es between the void ratio, e, underunsaturated conditions and the void ratio, es, under saturatedconditions at the same average skeleton stress state is aunique function of the bonding variable, �. This fundamentalassumption is successfully validated in this work by theanalysis of several published sets of experimental data fordifferent materials. The analysis of one set of data (forwhich both isotropic and shearing tests are available) alsosuggests that the relationship between e=es and � is uniquefor a given soil, and that it is independent of the appliedstress ratio.

On the basis of this assumption a full elasto-plasticstress–strain model for isotropic stress states is formulated,and its good performance is demonstrated by the comparisonbetween predicted and laboratory tests results from a com-prehensive experimental programme including a wide varietyof different stress paths. This comparison confirms thepotential of the proposed model for correctly predicting themost important features of the mechanical behaviour ofunsaturated soils by retaining at the same time the simplicityof a model formulated in terms of a single yield curve. Inparticular it is able to predict correctly the following twotypical responses of unsaturated soils that are not modelledby existing elasto-plastic constitutive frameworks based on asingle yield surface:

(a) the irreversible change of void ratio during drying(b) the dependence of the response during virgin compres-

sion at constant suction on the previous history ofsuction variation.

An additional significant advantage is that a reduced numberof laboratory tests are necessary for calibrating the proposedmodel. In particular, the relationship between e=es and � isthe only additional information required for the unsaturatedsoil behaviour (apart from the parameter values for thesaturated model). To define the relationship between e=es

and �, it is possible to choose among alternative testingoptions that involve irreversible straining of the soil such asvirgin loading at constant suction, undrained virgin loadingand wetting–drying at constant applied stress.

ACKNOWLEDGEMENTSThe authors wish to acknowledge the support of the

European Commission via a Marie Curie Fellowshipawarded to Dr Domenico Gallipoli and of the Ministerio deCiencia y Tecnologıa through research grant BTE2001–2227.

Dr Radhey Sharma carried out the experimental work

1.4

1.3

1.2

1.1

1

0.9

e

5 10 20 30 50 100 200 300 500p: kPa

(a)

Experimental

Prediction

Loading without wetting–drying

Loading after wetting–drying

(A)

(B)

0.6

0.4

0.2

0

�

Initial yield locus

Expanded yield locus

Loading without wetting–drying

Loading after wetting–drying

(A)

(B)

Drying

Wetting

5 10 100 1000p″: kPa

(b)

Fig. 17. Model prediction for isotropic virgin loading atconstant suction of 300 kPa (experimental data by Sharma,1998): (a) change of void ratio; (b) stress path

134 GALLIPOLI, GENS, SHARMA AND VAUNAT

described in this paper while he was a research studentat the University of Oxford, UK. The financial support ofthe EPSRC to such experimental programme (via grant noGR/J70512 awarded to Professor Simon Wheeler) is grate-fully acknowledged.

The authors thank Dr David Toll of the University ofDurham, UK, and Professor Simon Wheeler of the Univer-sity of Glasgow, UK, for useful discussions.

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ELASTO-PLASTIC MODEL FOR UNSATURATED SOIL 135