effect of degree of saturation on mechanical behaviour of unsaturated soils and its elastoplastic...

Computers and Geotechnics 37 (2010) 678–688

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Computers and Geotechnics

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Effect of degree of saturation on mechanical behaviour of unsaturated soilsand its elastoplastic simulation

De’an Sun *, Wenjing Sun, Li XiangDepartment of Civil Engineering, Shanghai University, 149 Yanchang Road, Shanghai 200072, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 September 2008Received in revised form 17 April 2010Accepted 17 April 2010Available online 14 May 2010

Keywords:Unsaturated soilsStress-state variableDegree of saturationTriaxial testElasto-plastic modelStress–strain behaviourCoupled model

0266-352X/$ - see front matter � 2010 Elsevier Ltd.doi:10.1016/j.compgeo.2010.04.006

* Corresponding author. Tel.: +86 21 56334259; faxE-mail address: [email protected] (D.A. Sun).

The two stress-state variable approach has been widely used in interpreting unsaturated soil behaviour.However this approach cannot take into account the effect of degree of saturation or water contents onthe stress–strain behaviour and strength of unsaturated soils. The triaxial test results presented in thispaper show that even if the same path of net stress and suction is followed, the stress–strain relationand strength are different due to different degrees of saturation. When other conditions are the same,the higher the degree of saturation for the soil sample is, the higher the stress ratio corresponding to agiven axial strain will be. This effect can be modeled by using an elasto-plastic constitutive model cou-pling hydraulic and mechanical behaviour of unsaturated soils. Comparisons between the predictedand measured results are presented, which demonstrate that the model can quantitatively simulatethe influence of the degree of saturation on stress–strain behaviour and strength of unsaturated soils.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Bishop [2] proposed a well known effective stress expression forunsaturated soils extending Terzaghi’s effective stress principle forsaturated soil. In Bishop’s expression, a concept of the equivalentpore pressure was introduced to consider the influence of thepore-air pressure ua and pore-water pressure uw on the stress–strain behaviour and strength of unsaturated soils. Jennings andBurland [9] indicated that the effective stress proposed by Bishopcould not explain the collapse phenomenon induced by wettingof unsaturated soils. So the stress–strain behaviour and strengthof unsaturated soils could not be described accurately by onlythe effective stress.

Fredlund and Morgenstern [5] demonstrated that the net stress(the difference between total stress and pore-air pressure) and thesuction (the difference between pore-air pressure and pore-waterpressure) could be considered as two stress-state variables to con-trol the mechanical response of unsaturated soils. Since then, anumber of non-linear elastic constitutive models have been devel-oped based on this double stress-state variables theory (e.g.,[6,11]). In 1990, an elasto-plastic model for unsaturated soils usingthe net stress and suction as stress-state variables was proposed byAlonso et al. [1]. The novelty in this model is the concept of theloading-collapse yield curve which can predict the collapse phe-

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nomenon of unsaturated soils. Subsequently several elasto-plasticconstitutive models for unsaturated soils have been developedwhich are based on experimental and theoretical studies[4,10,21]. Sun et al. [14] proposed a three-dimensional elasto-plas-tic constitutive model for unsaturated soils and used the experi-mental results obtained from a series of suction-controlledtriaxial compression and extension tests to verify the model, whichcould indeed reproduce the mechanical behaviour of the soilincluding collapse deformation under a three-dimension stressstate.

The above mentioned elasto-plastic constitutive models forunsaturated soils with two stress-state variables are limited tothe prediction of the mechanical behaviour of unsaturated soils.They did not take into account directly the effect of degree of sat-uration or water content on the stress–strain behaviour andstrength of unsaturated soils. In fact, the hydraulic (e.g., degreeof saturation) and mechanical (e.g., deformation and strength)changes take place simultaneously in unsaturated soils underexternal forces. Therefore, the current elasto-plastic constitutivemodel and soil–water characteristic curve model cannot predictsimultaneously the hydraulic and mechanical behaviour of unsat-urated soils.

In recent years, a small number of researchers have started topropose constitutive models to predict the hydraulic and mechan-ical behaviour of unsaturated soils using the elasto-plasticityapproach [7,12,13,19,20]. However, most of these models areformulated merely for isotropic stress conditions and can only

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describe qualitatively the hydraulic and stress–strain behaviour.Recently, Sun and Sheng [16] and Sun et al. [17] have proposedan elasto-plastic constitutive model coupling the hydraulic andmechanical behaviour under isotropic and triaxial stress condi-tions and used the experimental results obtained from suction-controlled isotropic and triaxial compression tests to verify themodel.

This paper presents firstly the results of triaxial tests on unsat-urated compacted specimens with different initial water contentsunder the same loading (net stress and suction) path, to demon-strate the effect of degree of saturation on the stress–strain behav-iour and strength of unsaturated soils. Secondly, the elasto-plasticmodel coupling hydraulic and mechanical behaviour proposed bySun and Sheng [16] and Sun et al. [17] is reviewed briefly. Finally,the model is used to predict the hydraulic and stress–strain behav-iour of unsaturated compacted soils with different initial degreesof saturation before compression and shearing. The results indicatethat the model can predict well the effect of the degree of satura-tion on the stress–strain behaviour and strength of unsaturatedsoils.

2. Triaxial testing program

A triaxial testing program is designed with the main objectiveto investigate the influence of the degree of saturation on themechanical behaviour of unsaturated soils.

2.1. Triaxial apparatus for testing unsaturated soils

Fig. 1 shows schematically the triaxial apparatus used in this re-search. The triaxial apparatus can control and measure the matricsuction s by using the axis-translation technique. The radial dis-placement of the specimens is measured using three rings madeof bronze mounted at H/4, H/2 and H from the top of the specimen,where H is the height of the specimen [15]. One of these rings wasput near the bottom of the specimen because the radial displace-ment at H was not completely zero although there is friction be-tween the specimen and pedestal. The specimen radial strain canbe calculated from the measured radial displacement, using the

Fig. 1. Triaxial apparatus for testing unsaturated soils.

assumption that the lateral shape is approximated by a third-orderpolynomial.

The pore-water pressure is maintained at atmospheric pressureduring the constant suction and wetting tests through a ceramicdisk installed in the pedestal with an air-entry value of 300 kPa.The pore-air pressure is applied at the top cap through a polyflonfilter that can prevent water within specimens from seeping outof the upper porous stone. Hence, the change in the water volumein the burette is the same as the change in the water volume of thetested specimen, thus allowing the degree of saturation to be mea-sured. More details can be found in Sun et al. [15].

2.2. Testing material and specimen preparation

The soil used in this study is called Pearl clay. It consists of 50%silt and 50% clay with the grain diameter less than 5 lm, and has aliquid limit of 49%, a plasticity index of 22% and a specific gravity of2.71. The composition of the clay, as determined using X-ray dif-fraction, is quartz, pyrophyllite, and kaolinite in order of domi-nance. There are few expansive clay minerals in Pearl clay. Thesoil was firstly air-dried and then mixed with the required amountof water to reach the specified water content. The soil sample waskept in polyethylene bags and stored in a constant temperatureand humidity room for several days to reach equalization. Triaxialspecimens, 35 mm in diameter and 80 mm high, were prepared bycompaction in a steel mold. Specimens were compacted in five lay-ers, and each layer was statically compacted using a 12 mm diam-eter plunger up to a vertical stress of 300 kPa. The number of blowsper layer varying from 12 to 18 resulted in a void ratio rangingfrom 1.20 to 1.31. The initial state of four specimens after compac-tion is summarised in Table 1. A detailed description of the com-paction procedure and the compaction curves can be found inSun et al. [18]. All the tested specimens are on the dry side ofthe optimum and the water content was about 8–10% less thanthe optimum value. The initial suction was about 110 kPa. The ini-tial suction was measured by the pore-water pressure transducerwith an applied pore-air pressure of 100 kPa under undrainedcondition.

2.3. Testing procedures

A total of four isotropic and triaxial compression tests were per-formed on unsaturated compacted specimens. Two of them wereperformed at a constant suction of 100 kPa, and the other two ata suction of 150 kPa. The stress paths imposed on the four speci-mens are shown in Fig. 2 – in which point A represents the initialstate of four specimens, and p and q are the mean net stress and thedeviator stress respectively. To investigate the influence of degreeof saturation on the stress–strain behaviour and strength of unsat-urated soils, the following two stress paths were applied during thetests: (i) a suction of 100 kPa or 150 kPa was imposed directly onspecimens under the isotropic net stress of 20 kPa (Test 1 and Test2); and (ii) after the suction of 100 kPa was firstly applied on thetwo specimens under the isotropic net stress of 20 kPa, the isotro-pic net stress was increased to 50 kPa under constant suction, andthen the specimens were wetted by decreasing the imposed suc-tion step by step from 100 kPa to 0 kPa under the isotropic netstress of 50 kPa. Finally the imposed suction was increased, alwaysunder an isotropic net stress of 50 kPa, to 100 kPa or 150 kPa forthe two specimens respectively (Test 3 and Test 4). After complet-ing the above paths the four specimens were all compressed to amean net stress of 200 kPa step by step under constant suctions.Finally specimens were sheared to failure under mean net stressof 200 kPa and suction of 100 kPa or 150 kPa in triaxial compres-sion stress. The stress–strain relation and the change in water con-tent were recorded throughout the above procedure. The degrees

Table 1Initial state of specimens after compaction.

Initial water content w0 (%) Dry density after compaction qd0 (g/cm3) Void ratio after compaction e0 Initial degree of saturation Sr0 (%)

Test 1 25.5 1.230 1.20 57.4Test 2 25.8 1.196 1.27 55.0Test 3 25.8 1.199 1.26 55.6Test 4 24.3 1.169 1.31 50.1

Fig. 2. Stress path: (a) in p � q � s space and (b) in p � s plane.

80

90

100

E

Drying D'

a

680 D.A. Sun et al. / Computers and Geotechnics 37 (2010) 678–688

of saturation in Test 3 and Test 4 experiencing the wetting processare different with those in Test 1 and Test 2. Thus, the stress–strainbehaviour and strength of unsaturated soils can be compared atdifferent degrees of saturation along the same stress and suctionpaths.

1 5 10 50 10040

50

60

70

s(kPa)

Sr(

%)

Test 1 Test 4

F'

F

Wetting

DA4B'

A1

1 5 10 50 10050

60

70

80

90

100

s(kPa)

Sr(

%)

Test 2 Test 3

D

E

D'Wetting

Drying

b

A2B'A3

Fig. 3. Variations in suction and degree of saturation from initial to (a) point F andF0 and (b) point D and D0 in Fig. 2.

2.4. Test results

Figs. 3 and 4 show the variations in the degree of saturation andvoid ratio from the initial state (i. e., after compaction, point A inFig. 2), to the same state point of four tests with isotropic net stressof 50 kPa and suction of 100 kPa (point D or D0 in Fig. 2) or 150 kPa(point F or F0 in Fig. 2). Points A1, A2, A3, and A4 denote the initialstates of the specimens for Test 1, Teat 2, Test 3 and Test 4 respec-tively. The degrees of saturation at point D0 and F0 in Test 3 and Test4 which experienced the wetting–drying process are obviouslyhigher than those at points F and D in Test 1 and Test 2, respec-tively. From Fig. 4, it can be seen that void ratio at point F of Test1 is almost the same as that at point F0 of Test 4 (see Fig. 4a) whilevoid ratio at point D of Test 2 is a little greater than that at point D0

of Test 3 (see Fig. 4b). Thus, tests on the specimens with almost thesame void ratio and different degrees of saturation are conductedby imposing the same loading path of net stress and suction, toinvestigate the influence of the degree of saturation on thestress–strain relation and strength of unsaturated soils.

Figs. 5 and 6 show the variations in degree of saturation andvoid ratio respectively during isotropic compression (net stress in-creases from 50 kPa to 200 kPa) under a constant suction of100 kPa or 150 kPa. As can be seen from Fig. 5, the degree of satu-ration increases with the mean net stress during isotropic com-pression even under a constant suction. From Fig. 6 it can beseen that the void ratios in Test 2 and Test 3 are quite close undera net stress of 200 kPa and suction of 100 kPa (Fig. 6a), and the voidratios in Test 1 and Test 4 are almost the same under a net stress of200 kPa and suction of 150 kPa (Fig. 6b).

Figs. 7 and 8 show the stress–strain behaviour and variations indegree of saturation during shearing under a constant mean net

stress of 200 kPa and a constant suction of 100 kPa (Fig. 7) and150 kPa (Fig. 8) respectively. It can be seen that the higher the sat-uration degree, the higher the stress ratio corresponding to a givenaxial strain and the higher strength when other conditions are thesame. This clearly illustrates that in addition to suction, the degree

0 100 2001.1

1.15

1.2

1.25

1.3

s(kPa)

Wetting

Drying

Test 2Test 3

b

D

E

D'

A3B'

A2

0 100 2001.1

1.15

1.2

1.25

1.3

1.35

s(kPa)

ee

Wetting

Drying

Test 1 Test 4

F

D

E

F'

aB' A4

A1 B

Fig. 4. Variation in void ratio from initial to (a) point F and F0 and (b) point D and D0

in Fig. 2.

0 100 20050

60

70

80

p(kPa)

S r(%

)

Test 3

Test 2

D' C'

D

C'

(a) s=100kPa

0 100 20050

60

70

S r(%

)

p(kPa)

Test 4

Test 1

F'C

(b) s=150kPa

FC

Fig. 5. Variations in degree of saturation during isotropic compression underconstant suction.


of saturation also affects the stress–strain relation and strength ofunsaturated soils.

These triaxial test results indicate that specimens with differentdegrees of saturation have different stress–strain behaviour andstrength along the same loading paths of net stress and suction.When other conditions are the same, the higher the saturation de-gree of the specimen, the higher the stress ratio corresponding to agiven axial strain and strength will be. Therefore, the stress–strainrelation and strength of unsaturated soils cannot be predicted justusing the net stress and suction as the stress-state variables with-out the degree of saturation being considered. The constitutivemodel for unsaturated soils should take into account the effectsof not only suction but also degree of saturation. Neither non-lin-ear elastic models (e.g. [6]) nor elasto-plastic models (e.g. [1]), for-mulated in terms of net stress and suction, can reflect thedifference in the stress–strain relation and strength of unsaturatedsoils due to the different degrees of saturation. However, more re-cent elasto-plastic constitutive models coupling the hydraulic andmechanical behaviour can reflect the influence of the saturationdegree on the stress–strain behaviour and strength of unsaturatedsoils by using the ‘average skeleton stress’ and suction as stress-state variables [13,16,20].

3. Hydro-mechanical elasto-plastic model

A hydro-mechanical elasto-plastic constitutive model for unsat-urated soils has been proposed by Sun and Sheng [16] and Sunet al. [17]. The model incorporates the influence of the degree ofsaturation on the stress–strain relation and strength, and the influ-ence of deformation on water-retention behaviour. It is applicableto unsaturated soils in which pore-air and pore-water are continu-

ous throughout the voids. In the following, the model is briefly re-viewed for predicting the above test results.

3.1. Stress-state variables for unsaturated soils

To identify the hydraulic and mechanical behaviour of unsatu-rated soils properly, the stress-state variables employed in themodel are the ‘average skeleton stress’ tensor r0ij and the suctions, and the strain state variables are the soil skeleton strain tensoreij and the degree of saturation Sr. The ‘average skeleton stress’ ten-sor r0ij is defined by

r0ij ¼ rij � uadij þ Srsdij ð1Þ

where rij is the total stress tensor, Sr is the degree of saturation, ua isthe pore-air pressure, and dij is the Kronecker delta. Eq. (1) is similarto the effective stress proposed by Bishop and Blight [3], with Sr tak-ing the place of the weighting factor v. The ‘average skeleton stress’automatically becomes Terzaghi’s effective stress for the saturatedstate.

3.2. Water-retention behaviour

In order to identify the water-retention behaviour of unsatu-rated soils, it necessary to know what factors affect mainly thebehaviour for a given soil. Fig. 9 shows some measured soil–watercharacteristic curves obtained by wetting tests on unsaturatedcompacted soils at almost the same density but different stressstates for each figure [18]. In Fig. 9, e0 and eb are the void ratiosat the state immediately after compaction (and before application

1

1.05

1.1

1.15

1.2

p(kPa)

(b) s=150kPa Test 1 Test 4

FF'

C

1021

1.1

1.2

1.3e

e

p(kPa)20010050

20010050

(a) s=100kPa Test 2 Test 3

D

D'

C'

Fig. 6. Variation in void ratio during isotropic compression under constant suction.

1

4

3

15-10ε 1(%)ε 3(%)

ε v(%)

ε (%)1

σ 1/σ 3

2

p=200kPas=100kPa

10

5

5-5

Test 2

Test 3

0 5 10 15

60

70

80

S r(%

)

Test 3

Test 2

Fig. 7. Stress–strain behaviour and variation in degree of saturation duringshearing under constant suction (p = 200 kPa, s = 100 kPa) with different initialdegrees of saturation.

ε (%)1

1

3

5

0

2010-10 ε 1(%)ε 3(%)

ε v(%)

σ1/σ3

5

p=200kPas=150kPa

Test 4

Test 1

0 10 2060

65

70

75

S r(%

)

Test 1

Test 4

Fig. 8. Stress–strain behaviour and variation in degree of saturation duringshearing under constant suction (p = 200 kPa, s = 150 kPa) with different initialdegrees of saturation.


of external stress) and at the beginning of the wetting tests respec-tively, R is the principal stress ratio (=r1/r3) in triaxial tests, Compand Ext denote triaxial compression and extension tests respec-

tively. It can be concluded from Fig. 9 that the soil–water charac-teristic curve is dependent on the density and is indirectlydependent on the stress state for a given soil. Thus, the void ratioshould be considered in mathematical modeling of the soil–watercharacteristic curve, in addition to the wetting process or dryingprocess. The soil–water characteristic relationship can be simplyidealized as shown in Fig. 10.

The main drying and main wetting equations of the soil–watercharacteristic relationship are expressed respectively by

Sr ¼ S0rdðeÞ � ksr ln s main drying

Sr ¼ S0rwðeÞ � ksr ln s main wetting

ð2Þ

while a scanning curve is given as:

Sr ¼ S0rsðeÞ � js ln s ð3Þ

In these equations, ksr and js are the slopes of the main drying (orwetting) curve and the scanning curve respectively; S0

rdðeÞ andS0

rwðeÞ are the degrees of saturation respectively on the main dryingand main wetting curves when s = 1, and are a function of the voidratio e, S0

rsðeÞ is the degree of saturation on the specific scanningcurve when s = 1, and is also a function of the void ratio e. The val-ues of S0

rdðeÞ, S0rwðeÞ and S0

rsðeÞ will depend on the void ratio and theunit chosen for the suction measurement. Throughout this paper itis assumed that the unit of suction is kilopascal (kPa).

According to the results of isotropic compression tests and tri-axial tests on unsaturated soils under constant suction, the void ra-tio versus degree of saturation relation under constant suction canbe approximated by Sun et al. [17]:

dS0riðeÞ ¼ �kse de; i ¼ d;w; s ð4Þ

where kse is the slope of the Sr � e curve under constant suction lar-ger than the air-entry value. kse may be dependent on suction [8],

1

λ sr

Sr

ln s

1

λ sr

Wetting Drying

κ se1e2

e1>e2

Fig. 10. Soil–water characteristic model at different void ratios.

0 p'

s

p0y

p'y

SI (s=sI)

SD(s=sD)

A B

CD

E

F

LC yield curve

Fig. 11. LC, SI and SD yield curves for isotropic stress states.

0 100 20050

60

70

80

90

100 e 0 e b R

1.28 1.18 1.0 1.42 1.20 1.5 Comp 1.31 1.16 1.5 Ext

(c) eba=1.18

S r(%

)

s(kPa)

0 100 20050

60

70

80

90

100 e 0 e b R

1.17 1.06 1.0 1.15 1.03 1.5 Comp

(a) eba=1.04

S r(%

)

s(kPa)

0 100 20050

60

70

80

90

100 e 0 e b R

1.28 1.12 2.0 Comp 1.21 1.10 2.0 Comp 1.21 1.12 2.0 Ext

(b)eba=1.11

Sr(%

)

s(kPa)

Fig. 9. Soil–water characteristic at similar densities but different stress states [18].


and but is here assumed to be a constant for simplicity. Differenti-ating Eqs. (2) and (3) and then combining them with Eq. (4) lead to:

dSr ¼ �kse de� ksrdss

ð5Þ

dSr ¼ �kse de� jsdss

ð6Þ

The above equations define the incremental water-retention behav-iour of unsaturated soils. In order to express the soil water charac-teristic curve simply, Eqs. (5) and (6) can be written as

dSr ¼ �kse de� bdss

ð7Þ

where

b ¼js for the scanning curveksr for the main wetting or drying curve

�ð8Þ

3.3. The constitutive model for isotropic stress states

A so-called ‘loading-collapse’ (LC) yield curve in the p0 � s planeunder an isotropic stress state is adopted in the following equation

p0y ¼ p0np0y

p0n

� �kð0Þ�jkðsÞ�j

ð9Þ

where p0y and p0y are the yield stresses for saturated soil and unsat-urated soils with suction s respectively (see Fig. 11); p0n is an isotro-pic stress at which no collapse occurs when the suction isdecreased; j is the swelling index in the e � ln p0 plane; and kð0Þand kðsÞ are the slopes of the normal compression lines of the satu-rated soil and the unsaturated soil with suction s in the e � ln p0

plane. Although the constant suction normal compression linesare not straight lines over the entire stress range in the e � ln p0

plane, they can be assumed to be linear over a limited stress range.The quantity kðsÞ is assumed as:

kðsÞ ¼ kð0Þ þ ksspa þ s

ð10Þ

where ks is a material parameter for identifying the change of kðsÞwith suction s.

From Eq. (9) we have

dp0y ¼@p0y@p0y

dp0y þ@p0y@s

ds ð11Þ

where

@p0y@p0y

¼ kð0Þ � jkðsÞ � j

p0y

p0n

� �kð0Þ�kðsÞkðsÞ�j

ð12Þ

@p0y@s¼

ksp0ypaðkð0Þ � jÞðkðsÞ � jÞ2ðpa þ sÞ2

lnp0np0y

!ð13Þ


When the stress state is inside the LC yield curve, the elastic volu-metric strain increment is given by

deev ¼

jdp0

ð1þ eÞp0 ð14Þ

The stress paths in the predictions of the ordinary unsaturatedsoil tests are specified in terms of net stress and suction. Incre-ments in p0 are given by differentiating Eq. (1) according to

dp0 ¼ dpþ Sr dsþ sdSr ð15Þ

where p is the mean net stress. Substituting Eqs. (7) and (14) intoEq. (15) gives

dp0 ¼ dpþ Sr dsþ s �kse de� bdss

� �

¼ dpþ ðSr � bÞdsþ ksejsdp0

p0ð16Þ

Rearranging Eq. (16) gives

dp0 ¼ dpþ ðSr � bÞds1� ksejs

p0ð17Þ

When the deformation is elastic, the increment in void ratio can becalculated by Eq. (14) with Eq. (17) using the increment in the netstress and suction, and the increment in the degree of saturationcan be calculated by Eq. (15), i.e.

dSr ¼ ðdp0 � dp� Sr dsÞ=s ð18Þ

When the stress state is on the LC yield curve, the plastic volu-metric strain increment is given by

depv ¼ðkð0Þ � jÞdp0y

ð1þ eÞp0yð19Þ

or, from Eq. (11),

depv ¼

kð0Þ � jð1þ eÞp0y

dp0y �@p0y@s

ds� �

@p0y@p0y

,ð20Þ

The stress paths in the predictions of the ordinary unsaturatedsoil tests are specified in terms of net stress and suction. Duringthe elastoplastic deformation the increments in p0y are given by dif-ferentiating Eq. (1) according to

dp0y ¼ dpy þ Sr dsþ sdSr ð21Þ

where py is the isotropic net stress in the elastoplastic range. Substi-tuting Eqs. (7), (14), and (20) into Eq. (21) gives

Fig. 12. Yield surface in p0 � q � s space.

dp0y ¼ dpy þ ðSr � bÞds� ksesde

¼ dpy þ ðSr � bÞdsþ ksesð1þ eÞðdeev þ dep

vÞ

¼ dpy þ ðSr � bÞdsþ ksesjp0y

dp0y

þ ksesfkð0Þ � jgp0y

dp0y �@p0y@s

ds� �

@p0y@p0y

,

¼ dpy þ Sr � b�ksesfkð0Þ � jg @p0y

@s

p0y@p0y@p0y

24

35ds

þ kseskð0Þ � j

p0y@p0y@p0y

þ jp0y

8<:

9=;dp0y ¼ dpy þ Bdsþ Aþ ksesj

p0y

!dp0y

ð22Þ

where

A ¼ ksesfkð0Þ � jgp0y

@p0y@p0y

ð23Þ

B ¼ Sr � b� A@p0y@s

ð24Þ

From Eq. (22),

dp0y ¼dpy þ Bds

1� A� ksesjp0y

ð25Þ

When the deformation is elastoplastic, the increment in void ratiocan be calculated by Eqs. (14) and (20) with Eq. (25) using the incre-ment in the net stress and suction, and the increment in the degreeof saturation can be calculated by Eq. (21), i.e.

dSr ¼ ðdp0y � dpy � Sr dsÞ=s ð26Þ

In addition to the LC yield curve, two more yield curves are usedto model hydraulic hysteresis as an elasto-plastic process, asshown in Fig. 11. The water-retention behaviour shown in Fig. 11is represented by a suction increase (SI) yield curve and a suctiondecrease (SD) yield curve in the p0 � s plane. When the suctionchanges during a drying (s P sI) or wetting (s 6 sD) process, thedegree of saturation increment is given by Eq. (5); otherwise, thedegree of saturation increment is given by Eq. (6). Therefore,according to the relation between the stress state (p0 and s) andthe yield curves (LC, SI and SD), different equations must be usedto calculate the strains and the degree of saturation. Whendp0y > 0 and s > sI or s < sD, the volumetric strains and the degreeof saturation are calculated by Eqs. (14), (19), and (5). Whendp0y > 0 and sD 6 s 6 sI, they are calculated by Eqs. (14), (19), and(6). When dp0y = 0, and s > sI or s < sD, the volumetric strain andthe degree of saturation are calculated by Eqs. (14) and (5). Whendp0y = 0 and sD 6 s 6 sI, they are calculated by Eqs. (14) and (6).

3.4. The constitutive model under axi-symmetric stress states

The elliptical shape is adopted as a yield curve in the p0 � qplane. Assuming an associated flow rule, the yield function (f)and the plastic potential function (g) are proposed to have the fol-lowing form:

f ¼ g ¼ q2 þM2p0ðp0 � p0yÞ ¼ 0 ð27Þ

1 5 10 50 10040

50

60

70

80

90

100

s(kPa)

S r(%

)

Meas. Pred.Test 1 Test 4

A1

B

E

F'F

Wetting

Drying

D

D'

A4B'

a

1 5 10 50 10050

60

70

80

90

100

s(kPa)S r(

%)


A3B'

D

E

D'Wetting

Drying b

A2

Fig. 13. Predicted and measured degrees of saturation during equalization,isotropic compression at constant suction, wetting and drying under isotropic netstress of 50 kPa.

0 100 2001.1

1.15

1.2

1.25

1.3

s(kPa)

Wetting

Drying


b

D

E

D'

A3A2

B'

0 100 2001.1

1.15

1.2

1.25

1.3

1.35

s(kPa)

ee

Wetting

Drying


F

D

E

F'

aB' A4

A1 B

Fig. 14. Predicted and measured void ratios during equalization, isotropic com-pression at constant suction, wetting and drying under isotropic net stress of50 kPa.


where M is the slope of the critical line in the p0 � q plane and q isthe deviatoric net stress. Fig. 12 shows the geometrical shape of theyield function in the p0 � q � s space.

The associated flow rule is obeyed in the ‘average skeletonstress’ space; that is

depij ¼ K

@f@r0ij

ð28Þ

where the proportionality constant K can be determined from theconsistency condition. Eq. (27) can be rewritten asf ¼ f ðp0; q; p0yÞ ¼ 0; leading to:

df ¼ @f@p0

dp0 þ @f@q

dqþ @f@p0y

dp0y ¼ 0 ð29Þ

The stress paths in the predictions of ordinary triaxial tests onunsaturated soils are specified in terms of net stress and suction.Increments in p0 are given by differentiating Eq. (1) according to

dp0 ¼ dpþ Sr dsþ sdSr ð30Þ

where p is the mean net stress in the elastoplastic range. Substitut-ing Eqs. (7), (14), and (20) into Eq. (30) gives

dp0 ¼ dpþ ðSr � bÞds� ksesde

¼ dpþ ðSr � bÞdsþ ksesð1þ eÞðdeev þ dep

vÞ

¼ dpþ ðSr � bÞdsþ ksesjp0

dp0


dp0y �@p0y@s

ds� �

@p0y@p0y

,

¼ dpþ Sr � b�ksesfkð0Þ � jg @p0y

@s

p0y@p0y@p0y

24

35ds


@p0y@p0y

dp0y þksesj

p0dp0 ¼ dpþ Bdsþ Adp0y þ C dp0

ð31Þ

where

C ¼ ksesjp0

ð32Þ

Rearranging Eq. (31) gives

dp0 ¼dpþ Bdsþ Adp0y

1� Cð33Þ

Substituting Eq. (33) into Eq. (29) and considering Eq. (11) gives:

df ¼ 11� C

@f@p0

dpþ @f@q

dqþ A1� C

@f@p0þ @f@p0y

!dp0y þ

B1� C

@f@p0

ds

¼ 11� C

@f@p0

dpþ @f@q

dqþ A1� C

@f@p0þ @f@p0y

!@p0y@p0y

dp0y

þ A1� C

@f@p0þ @f@p0y

!@p0y@sþ B

1� C@f@p0

" #ds ¼ 0 ð34Þ

where the isotropic yielding stress p0y for saturated soil is related tothe volumetric strain ep

v and is the same as that used in the tradi-tional Cam-clay model. Because the plastic volumetric strain ep

v isa hardening parameter in the model, the volumetric plastic strainsdep

v caused by dp0y in a saturated soil are the same as those in anunsaturated soils which are caused by dp0y and/or ds. Allowing forEq. (28), the following is obtained from Eq. (19):

dp0y ¼1þ e

kð0Þ � jp0y dep

v ¼1þ e

kð0Þ � jp0yK

@f@p0

ð35Þ


Substituting Eq. (35) into Eq. (34) and solving for K gives:

K ¼ �@f@p0 dpþ ð1� CÞ @f

@q dqþ A @f@p0 þ ð1� CÞ @f

@p0y

� �@p0y@s þ B @f

@p0

� �ds

A @f@p0 þ ð1� CÞ @f

@p0y

� �@p0y@p0y

p0y1þe

kð0Þ�j@f@p0

ð36Þ

From Eqs. (28) and (36), it is possible to calculate the plastic strainincrements caused by the increment in the net stress and/or thedecrement in suction.

4. Comparison of model predictions with experimental results

4.1. Model parameters and their determination

Because the proposed constitutive model is formulated withinan elasto-plastic framework, the strains consist of elastic and plas-tic components. The model requires five parameters to describe thestress–strain behaviour (kð0Þ; ks;j; p0n;M) and three parameters todescribe the water-retention behaviour (ksr ;js; kse).

To calculate the plastic strain, it is necessary to determine themodel parameters kð0Þ; ks;j; p0n and M. These model parametersare determined from the results of isotropic compression testswith wetting and loading–unloading–reloading processes, fol-lowed by subsequent triaxial compression tests on saturated andunsaturated soils under constant suction and constant p (or con-stant confining net stress). Two tests are here used to determinethe model parameters. First, an isotropic compression test is con-ducted on unsaturated soils under a constant suction. Secondly,

0 100 20050

60

70

80

p(kPa)

S r(%

)

Test 3

Test 2

D' C'

D

C'

(b) s=100kPa

0 100 20050

60

70

S r(%

)

p(kPa)

Test 4

Test 1

F'C

(a) s=150kPa

FC

Measured Predicted

Fig. 15. Predicted and measured degrees of saturation during isotropic compres-sion under constant suctions.

an unsaturated soils specimen is loaded to a small net stress(p = 20 kPa), and is then wetted to a saturated state for conductingan isotropic compression test with an unloading–reloading cycle,in order to obtain the saturated normal compression line and theinitial yield stress, p0y. A triaxial compression test was then con-ducted on this saturated specimen. From the result of the isotropiccompression test on the saturated specimen with an unloading–reloading stress path, kð0Þ and j can be determined. From theresult of the triaxial test on the saturated specimen, the internalfriction angle (/ or M) can be determined. The quantity p0n can bedetermined from the coordinates of the point where the twoisotropic compression lines for the saturated and unsaturatedspecimens intersect. ks can be found from the values of kð0Þ andthe compression index of the unsaturated specimen with suctions by using Eq. (10).

The model parameters ksr , js and kse define the water-retentioncurve and can be found as follows: kse is the slope of the e � Sr lineunder constant suction, so kse can be determined by plotting eagainst Sr for an isotropic compression test on unsaturated soilsunder a constant suction. Thereafter, js and ksr are determinedfrom the results of a wetting test at a small net stress for savingtest time by using Eqs. (2)–(4), or Eqs. (5) and (6). The elastic com-ponent is calculated from Hooke’s law. With Poisson’s ratio as-sumed to be 1/3, the Young’s modulus is calculated in the sameway as for the Cam-clay model:

E ¼ p0ð1þ eÞj

ð37Þ

The values of the relevant model parameters used in predictingthe stress–strain and water-retention behaviour of the Pearl clayare as follows:

1

1.05

1.1

1.15

1.2

ee

p(kPa)

50 100 200


F'F

C

(a) s=150kPa

1021

1.1

1.2

1.3

p(kPa)

20010050


D

D'

C'

(b) s=100kPa

Fig. 16. Predicted and measured void ratios during isotropic compression underconstant suctions.


kð0Þ ¼ 0:11;j ¼ 0:03; ks ¼ 0:12;p0n ¼ 1:6 MPa;M ¼ 1:15;kse ¼ 0:35; ksr ¼ 0:13;js ¼ 0:03

4.2. Model prediction versus experimental results

The above model is used to predict the measured unsaturatedsoil behaviour obtained from Test 1, Test 2, Test 3 and Test 4 de-scribed in Section 2.

Figs. 13 and 14 show the comparisons between predicted andmeasured degrees of saturation and void ratio during the equaliza-tion from point A to B or B0, isotropic compression from B0 to D orfrom B to F, and the wetting–drying tests with DED0 path for Test 3and DED0F0 path for Test 4. The measured results are the same asthose in Figs. 3 and 4, and the predicted results were obtainedusing the proposed coupled model for unsaturated soils with theabove model parameters. The initial (i.e., after compaction) valuesof yield stress (p0y) and suction (s0) used in all model predictionswere p0y = 25 kPa and s0 = 110 kPa. On the other hand, becausethe measured initial values of void ratios and degrees of saturationwere rather different between the four tests, these experimentalvalues were taken as the starting points of the respectivesimulations.

Figs. 15 and 16 show the comparisons of predicted and mea-sured results of the isotropic compression tests on unsaturatedcompacted Pearl clay under suctions of 100 kPa and 150 kPa. Itcan be seen from Figs. 15 and 16 that the model can predict notonly the deformation but also the change in the degree of satura-

1

3

-10

1(%)3(%)

v(%)

1/ 3

2

10

5

4

0

20

Meas. Pred.Test 2Test 3

a

Fig. 17. Predicted and measured results of triaxial compression tests for di

1

3

5

0

10-10 1(%)

3(%)

v(%)

1/ 3

5

4

5-5


a

Fig. 18. Predicted and measured results of triaxial compression tests for di

tion. From Fig. 15, we can see that the change in the degree of sat-uration during isotropic loading from 50 kPa to 200 kPa in Test 4and Test 3 are less large than those in Test 1 and Test 2, respec-tively. Generally, model predictions appear in agreement withthe test results.

Figs. 17 and 18 show comparisons of the predicted and mea-sured results of triaxial compression tests on unsaturated com-pacted Pearl clay under constant mean net stress (p = 200 kPa)and constant suctions (s = 100 kPa and s = 150 kPa). These testsare subsequent to the isotropic compression tests in Figs. 15and 16. It can be seen from Figs. 17 and 18 that the stress ratio(r1/r3)-strain relation and strength in Test 3 and Test 4 are high-er than those in Test 2 and Test 1, respectively. The model canpredict the differences due to different degrees of saturation forthe same imposed stress path and specimens with almost thesame density. It should pointed out that the Barcelona Basic Mod-el (BBM) cannot predict such differences because BBM uses thenet stress and suction as the stress-state variables without thedegree of saturation being directly incorporated. The predictedand measured results also show the degree of saturation in-creases with shearing.

It can be concluded from Fig. 13 to Fig 18 that the model withthe eight model parameters is capable of providing good predic-tions of the stress–strain and water-retention behaviour of unsat-urated compacted soil under isotropic and general stress paths.The effect of the saturation degree on the stress–strain behaviourand strength of unsaturated soils can also be taken into accountby the model.

0 5 10 15 20

60

70

80

90

S r(%

) Meas. Pred.Test 2Test 3

bε1(%)

fferent initial degrees of saturation under p = 200 kPa and s = 100 kPa.

0 5 10 15 2060

65

70

75

Sr(

%)

ε1(%)


b

fferent initial degrees of saturation under p = 200 kPa and s = 150 kPa.


5. Conclusions

(1) The results of triaxial tests on unsaturated soils indicate thatthe specimens with different degrees of saturation have dif-ferent stress–strain behaviour and strength even under thesame net stress and suction paths. The higher the degreeof saturation is, the higher the stress ratios correspondingto a given axial strain and the strength will be, when otherconditions are the same. This phenomenon cannot becompletely predicted by the models using the net stressand suction as the stress-state variables for unsaturated soilswithout taking into account the influence of degree ofsaturation.

(2) An hydro-mechanical constitutive model for unsaturatedsoils by using the ‘average skeleton stress’ and suction asthe stress-state variables is used to predict the stress–strain-strength and water-retention behaviour of unsatu-rated compacted specimens, with same stress state in termsof net stress and suction but different degrees of saturation,during isotropic and triaxial compression tests. The compar-isons between predicted and measured results indicate thatthe coupled model provides good description of the stress–strain and water-retention behaviour of unsaturated com-pacted clay during isotropic and triaxial compression tests,and the model can quantitatively simulate the effect ofdegree of saturation on the mechanical behaviour of unsatu-rated soils.

Acknowledgments

This research was sponsored by the National Natural ScienceFoundation of China (Grant No. 10972130) and the InnovativeFoundation for graduate students at Shanghai University, China.The authors also wish to thank the anonymous reviewer for hisconstructive and helpful comments and Professor WenxiongHuang at Hohai University, China and Dr. Jie Li at RMIT University,Australia for their kindly improving the paper.

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