Computers and Geotechnics - University of Newcastle · and Jommi [97] and D’Onza et al. [24] for anisotropy; Modaressi and Modaressi [77] and Cui et al. [18] for non-isothermal

Computers and Geotechnics 38 (2011) 757–776

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Computers and Geotechnics

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Review

Review of fundamental principles in modelling unsaturated soil behaviour

Daichao Sheng ⇑Centre for Geotechnical and Materials Modelling, School of Engineering, The University of Newcastle, NSW 2308, Australia

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

Article history:Received 5 February 2011Received in revised form 18 April 2011Accepted 9 May 2011

Keywords:Unsaturated soilsConstitutive modellingVolume changeShear strengthYield stressWater retentionHydro-mechanical coupling

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An unsaturated soil is a state of the soil. All soils can be partially saturated with water. Therefore, consti-tutive models for soils should ideally represent the soil behaviour over entire ranges of possible porepressure and stress values and allow arbitrary stress and hydraulic paths within these ranges. The lasttwo decades or so have seen significant advances in modelling unsaturated soil behaviour. This paperpresents a review of constitutive models for unsaturated soils. In particular, it focuses on the fundamentalprinciples that govern the volume change, shear strength, yield stress, water retention and hydro-mechanical coupling. Alternative forms of these principles are critically examined in terms of their pre-dictive capacity for experimental data, the consistency between these principles and the continuitybetween saturated and unsaturated states.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7572. Net stress and suction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7583. Volume change behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759

3.1. Separate stress and suction approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7593.2. Combined stress–suction approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7603.3. SFG approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7623.4. Stress-path dependent elastic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763

4. Yield stress versus suction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763
4.1. Relationship between yield stress and volume change equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7634.2. Reconstituted soils versus compacted soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766
5. Shear strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7686. Water retention behaviour and hydro-mechanical coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7697. Finite element implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7728. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773

1. Introduction

Soils that are partially saturated with water are often referred toas unsaturated soils. Some soils exhibit distinctive volume,

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strength and hydraulic properties when become unsaturated. Forthese soils, a change in the degree of saturation can cause signifi-cant changes in volume, shear strength and hydraulic properties.Nevertheless, the distinctive volume, strength and hydraulicbehaviour for unsaturated states should be treated as materialnonlinearity and modelled consistently and coherently. In otherwords, a constitutive model for a soil should represent the soil

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758 D. Sheng / Computers and Geotechnics 38 (2011) 757–776

behaviour over entire ranges of possible pore pressure and stressvalues and should allow arbitrary stress and hydraulic paths with-in these ranges. After all, any soil can be partially saturated withwater and an unsaturated soil is only a state of the soil, not anew soil, as pointed out by Gens et al. [36].

Soil mechanics principles are more established for soils at satu-rated states. Generalisation of these principles to unsaturated soilsrequires careful consideration of these fundamental issues: (1) vol-ume change behaviour associated with suction or saturationchanges, (2) shear strength behaviour associated with suction orsaturation changes, and (3) hydraulic behaviour associated withsuction or saturation changes. Soils can experience significant vol-ume changes upon changes of the degree of saturation or suction.Some soils expand upon wetting, some collapse and some do bothdepending on the stress level. The large volume changes associatedwith saturation change can lead to severe damages to foundationsand structures. Shear strength of soils can also change dramaticallyas the degree of saturation changes, and a related engineeringproblem is slope failures caused by rainfall. Unsaturated soils alsohave distinctive hydraulic behaviour which has profound implica-tions in designing cover and containment systems for variousindustrial and municipal wastes. These fundamental issues are in-deed the main concerns of unsaturated soil mechanics and its engi-neering applications [26,45].

Constitutive modelling of unsaturated soils generally involvesthe generalisation of constitutive models for saturated states tounsaturated states, by incorporating the fundamental issues men-tioned above. Research in this direction was pioneered by Alonsoet al. [2] and it has since attracted extensive interest. A large num-ber of constitutive models can now be found in the literature.There are several state-of-the-art reviews over the last 15 yearsor so, e.g. Gens [32], Wheeler and Karube [148], Kohgo [58], Wheel-er [146], Gens et al. [37], Sheng and Fredlund [107], Sheng et al.[110], Gens [33], Cui and Sun [19] and Gens [34]. These papersmay serve as good references for studying the topic. They usuallyprovide (1) a thorough discussion of stress state or constitutivevariables used to establish various models, (2) an in-depth analysisof specific constitutive models and their advantages and disadvan-tages, and (3) latest developments in the area of unsaturated soilmodelling. The papers by Gens [33,34] also provide interesting dis-cussions on the physical significance of different suction compo-nents and their roles in constitutive modelling.

Instead of a comprehensive review of existing constitutive mod-els for unsaturated soils, this paper devotes its main attention to anumber of specific issues: (1) volume change behaviour, (2) varia-tion of yield stress and shear strength with suction, (3) waterretention behaviour and hydro-mechanical coupling. These issuesrepresent the most fundamental components of constitutive mod-els for unsaturated soils. Alternative methods for tackling thesefundamental issues are scrutinised, particularly against the princi-ple that partial saturation is a state of soil and against their predic-tive capacity for experimental data. The issue of stress statevariables, a seemingly unavoidable topic in unsaturated soilmechanics, is not specifically discussed in this paper. In otherwords, constitutive models are not judged based on the stress statevariables they use, rather on their qualitative predictions of ob-served soil behaviour. In addition, a number of advanced topicsare not discussed in this paper:

Thermodynamics of unsaturated soils. This topic often leads tointeresting discussion of thermodynamically consistent stress-state variables and constitutive models. Interested readers may re-fer to, e.g. Houlsby [44], Hutter et al. [47], Gray and Schrefler [41],Sheng et al. [114], Li [63], Samat et al. [100], Coussy et al. [16] andZhao et al. [157].

Micromechanical modelling of unsaturated soils. This approachcan often lead to insights into soil behaviour and sometimes also

validation of macroscopic (continuum) constitutive equations.Some good examples of this approach include the work by Giliand Alonso [39], Jiang et al. [49], Katti et al. [52] and Scholtèset al. [104]. A closely related approach is the micro–macro dou-ble-structure models by Gens and Alonso [35], Alonso et al. [4],Sánchez et al. [101] and Cardoso and Alonso [13].

Thermo-hydro-chemo-mechanical modelling of unsaturatedsoils. In this approach, additional environmental variables suchas temperature and chemical concentrations are introduced tostudy soil behaviour. Some recent work in this area refers to Loretet al. [67], Guimarãs et al. [43], Cleall et al. [15], Kimoto et al. [56],Gens et al. [36,37] and Gens [34].

Miscellaneous topics such as non-isothermal behaviour, anisot-ropy, rate-dependent behaviour, degradation and damage, lique-faction of unsaturated soils. These topics are usually related tospecific soils or specific problems. Some representative work onthese topics are Cui and Delage [17], Stropeit et al. [121], Romeroand Jommi [97] and D’Onza et al. [24] for anisotropy; Modaressiand Modaressi [77] and Cui et al. [18] for non-isothermal behav-iour; Cardoso and Alonso [13] for degradation modelling; Arsonand Gatmiri [5] and Yang et al. [151] for damage modelling; Olde-cop and Alonso [85] and Pereira and de Gennaro [91] for rate-dependent behaviour; Unno et al. [138] and Bian and Shahrour[7] for liquefaction.

The paper is organised as follows. It first presents a brief discus-sion of two basic concepts used in constitutive modelling, i.e. thenet stress and the matric suction. The fundamental issues on vol-ume change, yield stress, shear strength, water retention and hy-dro-mechanical coupling are then discussed. The paper finallyoutlines the challenges and possible solution strategies for imple-menting unsaturated soil models into the finite element method.

2. Net stress and suction

Net stress is commonly used in interpreting unsaturated soilbehaviour and in constitutive modelling. It is defined as

�rij ¼ rij � dijua ð1Þ

where �rij is the net stress tensor, rij the total stress tensor, dij theKronecker delta, and ua the pore air pressure. The net stress is oftenused to analyse laboratory data, particularly those based on theaxis-translation technique where the air pressure is not zero. It issometimes perceived to recover the effective stress when soils be-come saturated. Such a perception should however be avoided. Un-der natural ground conditions where the air pressure isatmospheric, the net stress is equivalent to the total stress. Indeed,we never use the net stress concept to describe the behaviour of drysand. In other words, the atmospheric pore air pressure should beconsidered zero. Net stress is different from total stress only whenthe air pressure is not atmospheric.

The concept of net stress can be useful in interpreting experi-mental data based on axis-translation technique, if the techniqueis indeed valid for applying suction (see discussions on its validityin e.g. [80,6]). In this case, the air pressure is used as a referencevalue for stress measures and the net stress is the total stress in ex-cess of the pore air pressure.

The soil suction in the literature of unsaturated soil modellingusually refers to the matric suction and is usually expressed as:

s ¼ ua � uw ð2Þ

where s is called the matric suction in soil physics terminology andalso called the matrix suction [6], and uw the pore water pressure.The matric suction is used interchangeably with the matric poten-tial in soil–water potential. The latter is a measurement of energyand consists of two parts: capillary and adsorptive potentials. When

1.5

2.0

2.5

3.0

1 10 100 1000

v

p, kPa

uw=0uw=-10 kPa

uw=-100 kPa

Air entry value larger than 100 kPa

Fig. 1. Normal compression lines for saturated clay under constant pore waterpressures (k = 0.2, N = 3).

D. Sheng / Computers and Geotechnics 38 (2011) 757–776 759

pore water exists as capillary water at relatively high degrees of sat-uration, the capillary potential is dominant in the matric potentialand the definition by (2) is then considered to be valid. However,when pore water exists as adsorbed water films at low degrees ofsaturation, the adsorptive potential (wa) becomes dominant in thematric potential. Consequently, questions have been raised regard-ing whether Eq. (2) is still valid for the matric potential [6]. In thecase when water exists as adsorbed films to solid particles, the truewater pressure is not well defined. It is not unique at one materialpoint and is dependent on the proximity to the solid particle surface[70]. However an apparent water pressure can be introduced:uw = ua � wa, i.e. the apparent water pressure represents the nega-tive adsorptive potential measured in excess of air pressure. Suchan apparent water pressure is then unique at one material point.With such a definition of uw, the matric potential can be expressedby Eq. (2) over the full range of saturation. Nevertheless, the matricsuction should be differentiated from capillary phenomena. Its twocomponents may not easily be separable in a soil with double-porosity. As pointed out in Gens [34], it is more appropriate to thinkof matric suction as a variable that expresses quantitatively the de-gree of attachment of water to solid particles that results from thegeneral solid/water/interface interaction.

In constitutive modelling, the matric suction is often treated asan additional variable in a stress space for establishing constitutivelaws. This approach was pioneered in the Barcelona Basic Model(BBM) by Alonso et al. [2] and is then followed in most existingmodels, with a few exceptions where the suction is treated as aninternal variable or hardening parameter (e.g. [9,68]). Again, theseapproaches are not differentiated in this paper. Since the matricsuction can vary independently of stress, it is treated as an inde-pendent axis in the stress space in this paper. In addition, the mat-ric suction is considered to coincide with the negative pore waterpressure for fully saturated states and thus the suction axis extendsfrom negative infinity to positive infinity in the stress space.

3. Volume change behaviour

The volume change behaviour is one of the most fundamentalproperties of soils. For unsaturated soils, the large volume changesassociated with suction change can cause severe damages to foun-dations and structures. The volume change equation also under-pins the yield stress–suction and shear strength–suctionrelationships [110]. It is indeed the only absolutely necessary com-ponent that is needed to extend a saturated soil model to unsatu-rated states. The model that defines the volume change caused bystress and suction changes should again be applicable to the entirerange of possible pore pressure or suction values. The discussionsbelow are limited to isotropic stress states. The volume changeassociated with changes of deviator or shear stress has to be con-sidered in a three-dimensional constitutive framework, which de-pends on the specific model used for saturated soils, and is outsidethe scope of this paper.

For saturated soils, a common starting point is the linear rela-tionship between the specific volume (v) and the logarithmic effec-tive mean stress (ln p0) for normally consolidated soils:

v ¼ N � k ln p0 ¼ N � k lnðp� uwÞ ð3Þ

where p is the mean stress, k is the slope of the v � ln p0 line, and Nis the intercept on the v axis when ln p0 = 0. Eq. (3) is only valid forpositive increments of the effective stress. For unloading andreloading, the volume change depends on the specific plasticityframework adopted in the constitutive model. For example, hypo-plasticity and bounding surface plasticity adopts different volumechange mechanisms than classical elastoplasticity. However, for

normally consolidated soils subject to positive stress increments,Eq. (3) is usually used independently of the theoretical framework.

It should be noted that Eq. (3) represents a straight line in thev � ln p space only if the pore water pressure is zero. If the porewater pressure were kept at a negative value (suction), equation(3) would predict a smooth curve in the v � ln p space, as shownby Fig. 1. The air entry suction for the soil in Fig. 1 is assumed tobe larger than 100 kPa, so that the soil remains saturated. Indeed,these compression lines look very much like those for overconsol-idated soils. However, the curvature of the normal compressionlines is purely due to the nature of the logarithmic function andthe translation from the effective stress space (v � ln p0) to the totalstress space (v � ln p), not due to overconsolidation.

Equation (3) can also be written in an incremental form asfollows:

dv ¼ �kdp

p� uw� k

dð�uwÞp� uw

ð4Þ

It is clear that a negative increment in pore water pressure has ex-actly the same effect on the volume change of a saturated soil as anequal positive increment in mean stress.

In the literature, Eq. (3) is extended to unsaturated states in oneof the three approaches:

� Approach A: Separate stress and suction approach, or the netstress and suction approach.� Approach B: Combined stress–suction approach, or the effective

stress approach.� Approach C: SFG approach, which is a middle ground between

Approach A and B.

These approaches are discussed separately below in terms oftheir advantages and disadvantages.

3.1. Separate stress and suction approach

In Approach A, the volume change due to stress change is sep-arated from that due to suction change. A typical example of thevolume change equations in this approach is:

v ¼ N � kvp ln �p� kvs lnsþ uat

uat

� �ð5Þ

where �p is the net mean stress, N is the specific volume whenln �p ¼ 0 and s = 0, kvp is the slope of an assumed v � ln �p line orthe compressibility due to stress change, kvs the slope of an as-sumed v � ln s line or the shrinkability due to suction change, and

λ (Sr=1)

λvp

λvs

Sr (%)

λ

Fig. 2. Variation of compressibilities with degree of saturation (after [136]).


uat is the atmospheric pressure and is added to avoid the singularitywhen s = 0. Again, Eq. (5) is only used for increasing mean stress orincreasing suction. Indeed, kvs is usually replaced by the elasticcompression index jvs in most applications, unless the suction in-creases to a historically high value (above the suction-increase yieldsurface, see section below on yield stress).

Eq. (5) has been used in many models such as Alonso et al. [2],Wheeler and Sivakumar [150], Cui and Delage [17], Chiu & Ng [14],Georgiadis et al. [38] and Thu et al. [134]. The main advantage ofEq. (5) is that the compressibility due to stress and suction changesare dealt with separately. This does not only provide extra flexibil-ity for modelling soil behaviour, but is also supported by experi-mental data. Toll [136] and Toll and Ong [137] showed that thetwo compressibilities kvp and kvs can be totally different (Fig. 2).It is usually true that the suction shrinkability (kvs) decreases withdecreasing degree of saturation. On the other hand, the stress com-pressibility (kvp) can increase with increasing suction, particularlyfor compacted soils where highly compressible macropores(inter-aggregate pores) are present [96,30].

However, there are also a few shortcomings about Eq. (5). First,Eq. (5) does not recover Eq. (3) when the soil becomes saturated. Itrepresents a linear v � ln �p relationship for constant suctions, un-less kvp is assumed to be a function of stress (as in [38]. This isnot consistent with the saturated soil model (Fig. 1). As a conse-quence, the volume change becomes undefined at the transitionsuction between saturated and unsaturated states. Second, the

E1 NC 25 kPa

E7 NC 200 kPa

E11 NC 400 kPa

E2 NC 50 kPa

E5 OC 200 kPa

E6 OC 800 kPa

1 10 100 1000 Suction (kPa)

Void

ratio

0.9

0.8

0.7

0.6

0.5

0.4

Fig. 3. Variation of shrinkability with stress (after [23]).

volume change caused by suction changes is independent of stress,which is at variance with experimental observation shown inFig. 3. In addition, the atmospheric pressure (uat) in (5) makesthe suction change insignificant when the suction is less than theatmospheric pressure (s < uat). The first point, i.e. the discontinuityat the transition between saturated and unsaturated states, wasalso one of the reasons that some researchers turned to the effec-tive stress approach (e.g. [112]). A simple numerical example willillustrate this problem. Let a soil be compressed at the transitionsuction (ssa) from mean stress 1 kPa to 100 kPa. Let the air pressureremain atmospheric. In the saturated zone, the volume changesaccording to Eq. (3):

Dvjs�sa¼ �kvp ln

100þ ssa

1þ ssa

In the unsaturated zone, the volume changes according to Eq. (5):

Dvjsþsa¼ �kvp ln 100

These two volume changes can be quite different, depending on thevalue of the transition suction. The transition suction is either theair-entry or the air-expulsion value, depending on the hydraulicpath.

3.2. Combined stress–suction approach

In Approach B, the matric suction and the net mean stress arecombined into one single variable, i.e. an effective stress, to definetheir effects on soil volume. A general form of the effective stress is

p0 ¼ �pþ f ðsÞ ð6Þ

where f is either a function of suction or a function of suction anddegree of saturation. Such a definition of effective stress is very gen-eral and covers most existing definitions in the literature (but per-haps not the recent one by [144]. It is noted that function f(s)usually involves material state variables such as the degree of satu-ration and the air entry value. As a consequence, the definition ofthe effective stress and hence the stress space can change withmaterial states, a feature which is not shared by its counterpartfor saturated soils.

With such an effective stress, Eq. (3) is usually assumed to bestill valid for unsaturated states:

v ¼ N � k ln p0 ¼ N � kðsÞ lnð�pþ f ðsÞÞ ð7Þ

where N is the specific volume when ln p0 = 0. If the effective stressis indeed effective in controlling soil volume, v should remain con-stant under constant p0. As such, parameters N and k should be inde-pendent of suction. However, this is seldom the case in mosteffective stress models. In the literature, k is usually assumed tobe function of s, while N is treated either as a constant or a variable.We first discuss the case where N is a constant and then show that avarying N with suction cannot be recommended.

Eq. (7) is widely used in the literature and in fact most effectivestress models adopt it as the volume change equation (e.g.[9,12,50,57,59,68,82,83,92,103,112,114,123,125]). Eq. (7) generallyrecovers Eq. (3) when the soil becomes saturated. This is one of thegreatest advantages of using the effective stress.

However, there are also some disadvantages with Eq. (7). Theobvious issue is the difficulty in addressing the different compress-bilities due to stress and suction changes, as shown in Fig. 2, sincethere is now only one compressibility in the volume change equa-tion. The second issue is related to a constraint on the compress-ibility k. Let a saturated slurry soil be dried from zero suction toan arbitrary suction under constant mean stress of 1 kPa, i.e. thestress path AB in Fig. 4. As will be shown in the section on yield

1 pr

N

ln p

A

B

C

s=0

s>0, ds=0

v

Drying path constant p

N

ln p′

A

B

s=0

s>0, ds=0

v

Drying path under constant p ?

C

(a) (b)

Fig. 4. Normal compression lines according to the effective stress approach – constraint on k(s) with a constant N: (a) net mean stress space, (b) effective mean stress space.


stress (Fig. 9), this drying path is elastoplastic, not purely elastic.The volume of the soil then changes according to:

vB ¼ N � kðsÞ lnð1þ f ðsÞÞ ð8Þ

Now compress the soil under constant suction, i.e. stress path BC inFig. 4. The compression line will be curved in the v � ln �p space, dueto the f(s) term. If the suction at point B is above the air entry value,this compression line is expected to intersect with the initial com-pression line for saturated states. Let the intersection be point C (atnet mean stress of �pr). The volume at C is then:

vC ¼ vB � kðsÞ ln�pr þ f ðsÞ1þ f ðsÞ

� �ð9Þ

Along path AC, the volume changes according to:

vC ¼ N � kð0Þ lnð�pr þ f ð0ÞÞ ¼ N � kð0Þ ln �pr ð10Þ

Replacing (8) into (9) and then equating (9) with (10) lead to:

kðsÞkð0Þ ¼

lnð�prÞlnð�pr þ f ðsÞÞ < 1 ð11Þ

We usually anticipate the effective stress to increase with increas-ing suction, at least at low suction values. Therefore, we have:k(s) < k (0), meaning that the slope of the compression line de-creases with increasing suction. Such a constraint on k is howevernot supported by experimental data. In Alonso et al. [2], the slopeof the compression lines decreases with increasing suction. In thedata by Toll [136], Sivakumar and Wheeler [117], Toll and Ong[137] and Gallipoli et al. [30] for compacted soils, the slope of thecompression lines increases with increasing suction. In addition,data on wetting-induced collapse (e.g. [126]) do not support an everincreasing collapse volume with increasing mean stress.

In some constitutive models that use the combined stress–suc-tion approach, parameter N is assumed to vary with suction. If Ndecreases with increasing suction, the same constraint on k, i.e.Eq. (11) would apply. To avoid this constraint, N has to increasewith increasing suction. However, the variation of N with suctionhas implications on the yield surface evolution. An increasing Nwith suction basically implies that drying a slurry soil under con-stant effective mean stress will cause the yield surface to expand,which is not consistent with the stress path. This inconsistency willbe discussed in association with Fig. 9 below.

Gallipoli et al. [30] proposed the following volume changeequation:

v ¼ ðN � k ln p0Þð1� að1� expðbnÞÞ ð12Þ

where N and k are the two parameters of the normal compressionline for saturated states, a and b fitting parameters, and n a positivevariable representing the bonding effects of suction. The bondingvariable (n) is a function of both s and Sr. Eq. (12) is hence equivalentto Eq. (7) with N and k being functions of s and Sr. Gallipoli et al.[29,30] showed that Eq. (12) is able to predict the volume changeat both normal compression and critical states for a variety of com-pacted soils. One challenge in using Eq. (12) is that the yield stress islikely to be functions of both suction and degree of saturation. Be-cause the s–Sr relationship is usually not unique due to hydraulichysteresis, the resulting loading-collapse surface may not be welldefined. However, Gallipoli (personal communication) suggests thatit is sufficient to define the bonding variable (n) in terms of Sr only(e.g. n = 1 � Sr), which would then resolve the non-uniquenessproblem and lead to a unique loading-collapse surface in the Sr � p0

space.To avoid the constraint defined by Eq. (11), a possible augmen-

tation to Eq. (7) is to assume that the compressibility k is a functionof degree of saturation, i.e. k(Sr), while keeping N constant. This issimilar to the approach by Gallipoli et al. [30] with n being a func-tion of Sr only. Because Sr changes with soil volume even if the suc-tion is kept constant, the slope of the compression line will thenchange and will most likely increase with increasing mean stress.Therefore, it is possible to have k(Sr) decreasing with decreasingSr, but the slope of the compression line for constant suction in-creases with increasing stress (Fig. 5b). Al-Badran and Schanz [1]used an approach similar to Fig. 5b, but formulated their volumechange equation in the net stress–suction space. Kikumoto et al.[55] and Zhang and Ikariya [152] assumed a saturation-dependentcompressibility (k(Sr)), but formulated their models in the effectivestress–suction space. The same issue of non-uniqueness of theyield surface in the stress–suction space will arise, due to hydraulichysteresis. To avoid this non-uniqueness problem, the yield surfaceand hence the shear strength can be defined in the Sr � p0 space,thus eliminating suction as the additional variable of the stressspace. Some work in this direction has recently been reported byZhou et al. [159].

Clearly further research is required if the combined stress–suction approach is adopted for the volume change, particularly interms of consistent explanation of the suction-caused and stress-caused volume changes for reconstituted soils. A worthwhileendeavour in this direction is perhaps to explore the possibilities

1ln p′

A

NCL(s>0, ds=0)

v

s=0

( )rSλ

NNCL under constant rS

1

N(s)

ln p′

A

NCL(s=0) NCL(s>0, ds=0)

v

Drying path under constant p′ ?

N(0)

( )r,s Sλ

Drying path under constant p ?

(a) (b)

Fig. 5. Other options of normal compression lines according to the effective stress approach: (a) N increases with increasing suction. (b) k is a function of degree of saturation(Sr).


of using k(Sr) in Eq. (7) and the Sr � p0 space to establish all consti-tutive equations.

3.3. SFG approach

Sheng et al. [108] proposed a third way to model the volumechange for unsaturated soils under isotropic stress states. Thenew model, referred to as the SFG model, represents a middleground between Approach A and Approach B and is expressed inan incremental form as follows:

dv ¼ �kvpd�p

�pþ f ðsÞ � kvsds

�pþ f ðsÞ ð13Þ

Eq. (13) is in the same form as Eq. (4). Similar to Approach A, Eq.(13) is defined in terms of net stress and suction and separatesthe compressibilities due to the two variables, i.e. kvp and kvs. Sim-ilar to Approach B, it combines the suction with the net mean stressin the denominator, i.e. the term �pþ f ðsÞ, and recovers Eq. (4) forsaturated states. The term �pþ f ðsÞ represents the interaction be-tween stress and suction and makes the normal compression linesfor non-zero suction curved in the v � ln �p space. However, thereis no constraint on parameter kvp. As a first approximation, kvp

can be assumed to be independent of suction, as indicated by thedata of Jennings and Burland [48] for air-dry soils. More realisticallyit should depend on suction. For example, the data of Sivakumarand Wheeler [117] shows that kvp increases with increasing suctionfor compacted soils. Parameter kvs must equal kvp when the soil isfully saturated, because of Eq. (4). It generally decreases withincreasing suction and approaches zero. Sheng et al. [108] sug-gested the following simple function for kvs

kvs ¼kvp; s 6 ssa

kvpssaþ1sþ1 ; s > ssa

(ð14Þ

where ssa is the transition suction and was also called the saturationsuction in Sheng et al. [108]. It is the unique transition suction be-tween saturated and unsaturated states in the SFG model.

We note that the number ‘1’ in Eq. (14) is used to avoid the sin-gularity when ssa = 0 and is not truly needed if ssa is not absolutelyzero. A better expression would be:

kvs ¼kvp; s 6 ssa

kvpssas ; s > ssa

(ð15Þ

The difference between Eqs. (14) and (15) is minimal, but Eq. (15) ispreferred. Eq. (15) can be applied as long as the transition suction isnot absolutely zero. Kurucuk et al. [61] used Eq. (13), but a differentfunction for kvs than (15). Again, both kvp and kvs can vary withstress path and take different values on a loading and unloadingpath respectively.

The function f(s) in Eq. (13) can also take different forms. Shenget al. [108] initially used the following function:

f ðsÞ ¼ s ð16Þ

This is perhaps the simplest form possible for f(s) and yet guaran-tees the continuity between saturated and unsaturated states. Evenwith this simplest form, Zhou and Sheng [158] showed that the SFGmodel is able to predict a good set of experimental data on volumechange and shear strength, both for soils reconstituted from slurryand for compacted soils. Due to the f(s) term in Eq. (13), the normalcompression lines will be curved in the v � ln �p space (Fig. 6). Fig. 6ashows the compression curves for a reconstituted soil under varioussuctions. The compression curves for s = 400, 650 1000 kPa are allnormal compression lines that do not involve any unloading orreloading. The SFG predictions for these curves were obtained withone single kvp value. Fig. 6a also shows the reconstituted soil be-comes stiffer as the suction increases. Fig. 6b shows the compres-sion curves for a compacted soil. The difference in the estimatedyield stresses by Eqs. (5) and (13) is clearly shown in the figurefor s = 300 kPa. The yield stress is indicated by the meeting pointsof the unloading–reloading line and the normal compression lines.The parameter kvp was allowed to change with suction in Fig. 6b.

One shortcoming of Eq. (16) is that the soil compressibility ap-proaches zero as suction increases to infinite. There is also a theo-retical discontinuity between unsaturated states and completelydry state (Sr = 0). To avoid these problems, an alternative form off(s) could be used, for example:

f ðsÞ ¼ Srs ð17Þ

This equation will not only guarantee the continuity between satu-rated and unsaturated states, but also the continuity betweenunsaturated and the completely dry state (Sr = 0). More interest-ingly, both Eqs. (16) and (17) will lead to the same shearstrength–suction relationship. The degree of saturation in (17) canalso be replaced by the effective degree of saturation (Se

r ), as sug-gested by Pereira and Alonso [90] when discussing Bishop’s vparameter. However, the performance of Eq. (17) is yet to be

v

(b) Compacted Kaolin(a) Air-dry silty clay

1 10 100 1000 10000

1.54

1.50

1.46

1.42

1.38

s = 0 kPa s = 400 kPa s = 650 kPa s =1000 kPa

Measured Predicted by eq. (13)

p (kPa)

Reloading line

NCL NCL

NCLs

p (kPa)

NCL fitted by eq. (13)

s = 0

s = 300 kPa

Reloading

2.25

2.15

2.05

1.95 10 100 1000

NCL fitted by eq. (5)

v

Reloading

Fig. 6. Predicted isotropic compression curves for (a) reconstituted silty clay (data by [20]), (b) compacted kaolin (data by [117]).


validated against experimental data, and the loading-collapse yieldsurface may become non-unique due to hydraulic hysteresis.

The SFG approach seems to be able to overcome some disadvan-tages of Approach A and Approach B. The main disadvantage of theSFG approach is that it exists only in an incremental form and itsintegration depends on stress path [154,109]. The stress pathdependency requires special treatment in the stress integrationof the constitutive model [111].

3.4. Stress-path dependent elastic behaviour

An interesting issue related to the volume change is that all theapproaches discussed above are stress-path dependent. Zhang andLytton [154] and Sheng et al. [109] showed that Approaches A andC lead to stress-path dependent elastic behaviour. Approach B canalso result in stress-path dependent elastic behaviour, because aclosed loop of net stress and suction changes do not necessarilylead to a closed loop of effective stress changes (Fig. 7). Suchbehaviour is caused by the material state dependency of the effec-tive stress. For example, the change of the effective mean stressalong path AB in Fig. 7 is usually different from that along pathCD, because the degree of saturation (or the air entry value) haschanged from A to C, even though the net mean stress is kept con-stant during both paths. Such an unclosed loop means that themodel is stress-path dependent, even if the stress changes occurinside the elastic zone. A model that exhibits stress-path depen-dent elastic behaviour is at variance with classical elastoplasticitytheory and thermodynamics, and hence should generally be

A

B

C

D

p

s s

A

B

C

D

p′

D

Fig. 7. A closed loop in the net stress space and its corresponding open loop in theeffective stress space.

avoided. However, it is still a challenge to develop a model thatexhibits a stress-path independent elastic behaviour.

4. Yield stress versus suction

4.1. Relationship between yield stress and volume change equation

In the literature of unsaturated soil mechanics, the yield stressof an unsaturated soil is usually assumed to be a function of suc-tion. The concept of yield stress in classical elastoplasticity theoryrefers to the stress level that causes plastic deformation. In otherplasticity frameworks such as bounding surface plasticity[21,99,78], hypoplasticity [60,72] and generalised plasticity[69,86,101], plastic deformation occurs along all loading pathsincluding reloading. In these cases, a loading function or a bound-ing surface is used to differentiate unloading from loading. In thediscussion below, the yield surface concept is based on the classicalelastoplasticity theory, but can be extended to the loading orbounding surface concepts.

Under isotropic stress states, the yield stress is also called thepreconsolidation stress. For unsaturated soils, the yield net meanstress, denoted here by �pc, is conventionally determined from iso-tropic compression curves obtained at constant suctions. Thesecompression curves are usually plotted in the space of void ratioversus logarithmic net mean stress. The initial portion of the curveis typically flatter than the ending portion of the curve, if the suc-tion is larger than zero. Such a curve is then approximated by twostraight lines, one representing the elastic unloading–reloadingline and the other the elastoplastic normal compression line. Theintersection point of the two straight lines is used to define thepreconsolidation (yield) stress (Fig. 8a). The yield stress so deter-mined is generally found to increase with increasing suction, irre-spective of samples prepared from slurry states or from compactedsoils, leading to the so-called loading-collapse yield surface that iswidely used in constitutive models for unsaturated soils (Fig. 8b).

The procedure outlined above for determining the yield stress isbased on the assumption that the e� ln �p relationship for normallyconsolidated soils at s > 0 is linear and may not be consistent withthe definition of yield stress. To demonstrate this inconsistency, weshould first realise that the isotropic compression curves shown inFig. 8 are typical of unsaturated soils reconstituted from slurry (e.g.

(b) Variation of yield stress with suction (a) Isotropic compression curves

ln pc1p c2p

c3p

0s =

s1

1 2 30 s s s< < <

c1 c2 c3p p p< <

s2 s3

e s

p

s1

s2

s3

c1p c2pc3p

LC yield surface

Fig. 8. Isotropic compression curves under constant suction (s) and derived yield stresses.


[48]) as well as of compacted soils. Because it is easier to under-stand the preconsolidation stress for a slurry soil that for a com-pacted soil, we use a slurry soil as an example here. Let usassume that the slurry soil has not been consolidated (with a zeropreconsolidation stress). The initial yield stress for the soil is thenzero (Point A in Fig. 9a). We also note that the effective stress forsaturated states is constant along the 135� line in the �p� s space.

s

p

Current yield surfaces

45o

A

B

C D

F E G 135o

(a) Net stress – suction space

Zer

o sh

ear

stre

ngth

line

c0p

v

A

B

C

D

ln p

NCL (s>0, ds=0)

NCL (s=0)

(c) Corresponding volume change in net stress space

N

Fig. 9. Evolution of the elastic zone during drying and compres

Drying the slurry soil to suction B under zero mean stress is similarto consolidating the soil to stress E under zero suction (Fig. 9a). In-deed, if the air entry value of the soil is larger than suction at B, thelength AB is exactly the same as AE, due to the effective stress prin-ciple for saturated soils. However if the air entry value is lowerthan suction at B, the length AB should generally be larger thanAE, because a suction increment is generally less effective than

45o

A

B

C D

F G Ep′

s

Current yield surfacesZ

ero

shea

r st

reng

th li

ne

H

c0p′

(d) Corresponding volume change in effective stress space

ln p′

A, H

NCL(s=0)

NCL(s>0, ds=0) with λ(Sr)

vDrying path under constant

net mean stress

C

D

B

(b) Effective stress – suction space

sion of a slurry soil (A ? B ? C: constant net mean stress).

pc0p

Suction-increase yield surfaces

Loading collapse yield surfaces

Apparent tensile strength surface

Thu et al. (2007a) Alonso et al. (1990) Delage and Graham (1995)

s

Fig. 10. Loading-collapse, suction-increase and apparent tensile strength surfacesin various models.


an equal stress increment in terms of consolidating the unsatu-rated soil. Once the soil becomes unsaturated, the yield stress doesnot necessarily change with suction along the 135� line (denotedby the dashed line). However, the new yield surface always gothrough the current stress point, as shown in Fig. 9a. The elasticzone expands as the suction increases from A to B. The stress points(e.g. B and C) are on the current yield surfaces. Let now the soil beisotropically compressed under the constant suction at Point C(stress path CD in Fig. 9a). The isotropic compression path (CD) isagain outside the initial elastic zone and the soil at point C is nor-mally consolidated. Therefore, the isotropic compression path (CD)as well as the drying path from B to C is elastoplastic and does notinvolve a purely elastic portion as Fig. 8 indicates, suggesting thatthe method for determining the yield stress in Fig. 8 be conceptu-ally inconsistent. Indeed, the suction-induced apparent consolida-tion effect should refer to the increase of the preconsolidationstress at zero suction (�pcð0Þmoves from E to F as suction increasesfrom B to C in Fig. 9a), not the preconsolidation stress at the cur-rent suction. We also note that as the slurry soil is dried under con-stant stress, the soil becomes stiffer. The slope of the compressionline at point C is much smaller than that at point A in Fig. 9c.

The same analysis can be done in the effective stress–suctionspace (Fig. 9b). In the p0 � s space, the zero shear strength line orthe apparent tensile strength surface is commonly assumed tobe: p0 = 0, i.e. the vertical line that goes through the origin of thespace. This line must also be the initial yield surface for a soil thathas not been consolidated, i.e. p0cð0Þ ¼ 0. For a saturated soil thathas a zero yield stress, i.e. a slurry, the size of the elastic zone re-mains zero as long as the effective mean stress remains zero, irre-spective of the pore water pressure values. When the soil becomesunsaturated, the yield surface will continue along the zero shearstrength line, if the effective mean stress remains zero. To keepthe effective mean stress zero, a tensile net mean stress has to beapplied to balance out the suction increase. As a consequence,the size of the elastic zone remains zero, which also reflects theeffective stress principle. Furthermore, if the yield surface doesnot collapse to the zero shear strength line when p0cð0Þ ¼ 0, therewould be plastic deformation for loading along the yield surface,and the shear strength as well as the yield stress of the soil wouldbecome non-uniquely defined.

In the effective stress–suction space (Fig. 9b), the stress path forsuction increase under constant net mean stress is initially inclinedto horizontal by 45� for saturated states. Therefore, the stress pathwill cross the current yield surface and the drying path AB for aslurry soil is elastoplastic, not purely elastic. Once the soil becomesunsaturated, the stress path will drift away from the 45� line andthe yield surface will also drift away from the vertical line (thedashed lines in Fig. 9b). Nevertheless, the current stress points A,B, C, and D stay on the current yield surfaces and the stress pathABCD causes elastoplastic volume change, in consistency withFig. 9a.

Fig. 9b can also be used to illustrate that parameter N in the vol-ume change equation, i.e. Eq. (7), cannot be a function of suction.This is because increasing suction under constant effective meanstress, i.e. stress path AH in Fig. 9b, does not lead to the expansionof the elastic zone. To obtain stress path AH in Fig. 9b, a tensile netmean stress has to be applied as suction increases, so that theeffective mean stress remains zero. Such a stress path correspondsto the so-called neutral loading and does not lead to yield surfaceexpansion. The expansion of the elastic zone is a prerequisite forshifting the normal compression lines towards higher specific vol-ume in v � ln p0 space (Fig. 5a). Therefore, parameter N in Eq. (7)cannot be a function of suction, if the zero shear strength line is as-sumed to be p0 = 0. The corresponding volume change along pathABCD can still be explained consistently if the soil compression in-dex (k) is assumed to be a function of degree of saturation, as

shown in Fig. 9d. Points A and H are then identical in the v � ln p0

space.In the literature, the yield stress variation with suction is a

rather confusing part. Many early models adopt three yield sur-faces in the net stress–suction space, namely the loading-collapseyield surface, the suction-increase yield surface and the apparenttensile strength surface. Fig. 10 shows some examples of thesemodels. The loading-collapse yield surface is used to model thevolume collapse when an unsaturated soil is first loaded underconstant suction and then wetted under constant stress. Thesuction-increase yield surface is used to capture the plastic volumechange when an unsaturated soil is dried to a historically high suc-tion. The apparent tensile strength surface defines the zero shearstrength or the apparent tensile strength due to suction increase.These yield surfaces are usually defined separately. For example,setting the preconsolidation stress (�pc0) to zero in the loading-collapse yield function does not recover the apparent tensilestrength function. The suction-increase yield surface is usuallyhorizontal or gently sloped (Fig. 10) and is not related to theloading-collapse surface or the apparent tensile strength surface.

Sheng et al. [108] showed that the loading-collapse surface, theapparent tensile strength surface and the suction-increase yieldsurface are related to each other. In the SFG model, the yieldstress–suction relationship, the apparent tensile strength–suctionrelationship and the shear strength–suction relationship are all de-rived from the volume change equation, i.e. Eq. (13). In this model,the yield stress for a slurry soil that has never been consolidated ordried varies with suction in a unique function. This function alsodefines the apparent tensile strength surface or the zero shearstrength surface in the stress–suction space (the curve throughpoint A in Fig. 11). The curve approaches the 45� line as the suctionbecomes zero or negative (positive pore water pressure). Dryingthis slurry soil under zero stress (stress path ABC) causes theexpansion of the yield surface to point C in Fig. 10. Therefore, thesuction-increase yielding is already included in the yield functionand there is no need to define a separate function. If the unsatu-rated soil at point C is then compressed under constant suction(stress path CD in Fig. 11), the yield surface will evolve to the load-ing-collapse surface that passes through point D in Fig. 11. Theyield surface in the stress space represents the contours of thehardening parameter, which is usually the plastic volumetricstrain. The stress path CD will change the initial shape of the yieldsurface, because the plastic volumetric strain along CD depends onthe suction level. The loading-collapse yield function recovers theapparent tensile strength function when the preconsolidation

s

p

Suction increase yield surface

Apparent tensile strength

Loading collapse yield surface

A

B

D

45o 45o 45o

C

c0p cn0p

Fig. 11. Yield stress variation with suction in the SFG model [108].


stress at zero suction (�pc0) is set to zero. All these yield surfaces arecontinuous and smooth in the stress–suction space. Models basedon effective stresses also have these features of the SFG model, asshown in Fig. 9b.

As pointed out by Wheeler and Karube [148] and shown bySheng et al. [108] and Zhang and Lytton [156], the apparent tensilestrength function, the suction-increase yield function and the load-ing-collapse yield function are all related to the volumetric modelthat defines the elastic and elastoplastic volume changes caused bystress and suction changes. If Approach A, i.e. Eq. (5), is adopted todescribe the volume change, the loading-collapse yield surface willtake the following form:

�pc ¼�pc0 � s; s 6 ssa

�pr�pc0�ssa

�pr

� �kvp ð0Þ�jkvp ðsÞ�j

; s > ssa

8<: ð18Þ

where �pc0 is the yield stress at zero suction, �pr a reference stress,and j the elastic compression index. The specific shape of this func-tion depends on the variation of the compressibility with suction,i.e. the function kvp(s). The suction-increase yield surface can alsobe derived from Eq. (5). If the shrinkability (kvs) is assumed to beindependent of stress, the suction-increase yield surface is simply:

s ¼ s0 ð19Þ

where s0 is the yield suction. Eq. (19) represents a horizontal line inthe stress–suction space. Because Eq. (5) is not defined at zero suc-tion and zero mean stress, the apparent tensile strength surfacecannot be derived from (18). A separate function is usuallyintroduced:

�p0 ¼ �ks ð20Þ

where k was assumed to be constant in Alonso et al. [2], but varywith suction in Georgiadis et al. [38].

If Approach B, i.e. Eq. (7), is adopted to describe the volumechange, the loading-collapse surface can then be written as:

p0c ¼p0c0; s 6 ssa

p0rp0c0pr

� �kð0Þ�jkðsÞ�j

; s > ssa

8<: ð21Þ

The suction-increase surface is the same as Eq. (19). The apparenttensile strength surface is recovered from (21) by setting �pc0 ¼ 0:

p00 ¼ 0 ð22Þ

which represents the vertical line going through the origin of thestress space.

If the SFG model, i.e. Eqs. (13) and (15), is adopted to describethe volume change, the yield stress takes the following form:

�pc ¼�pc0 � s; s 6 ssa

�pc0 � ssa � ssa ln sssa; s > ssa

(ð23Þ

The function defines the suction-increase surface and is indepen-dent of the specific function f(s) used in (13). The apparent tensilestrength function is also defined in (23) by setting �pc0 ¼ 0. The load-ing-collapse surface takes a very similar form:

�pc ¼�pcn0 � s; s 6 ssa

�pcn0�pc0

�pc0 þ f ðsÞ � ssa � ssa ln sssa

� �� f ðsÞ; s > ssa

(ð24Þ

where �pcn0 is the new yield stress at zero suction (Fig. 11). This func-tion is however dependent on the specific forms of f(s) used in (13).

The functions defined in (18)–(24) are all continuous over theentire ranges of possible suction or pore pressure values. However,functions (18) and (21) may not be smooth, dependent on func-tions kvp(s) and k(s), respectively. On the other hand, functions(23) and (24) are continuous and smooth. All these functions canbe incorporated into existing constitutive models for saturatedsoils. For example, if the modified Cam clay model is used for sat-urated soil behaviour, the yield function can be generalised tounsaturated states along the suction axis:

f ¼ q2 �M2ð�p� �p0Þð�pc � �pÞ � 0 ð25Þ

where f is the yield function in the stress space, q is the deviatorstress, M is the slope of the critical state line in q� �p space, and�p0 and �pc are defined above. Again, Eq. (25) is valid for all pore pres-sure and suction values.

4.2. Reconstituted soils versus compacted soils

A soil can become partially saturated with water in differentways. Two types of unsaturated soil samples are often used in lab-oratory: (1) dry soil powders are statically or dynamically com-pacted at specified water contents, (2) samples reconstitutedfrom slurry are dried to unsaturated states. The first type of sam-ples (compacted soils) is far more common than the second typeof samples (reconstituted soils), because it is more difficult todesaturate a slurry sample than a compacted sample. It has beennoted that most constitutive models for unsaturated soils arebased on experimental data for compacted soils [108]. Compactedsoil samples can be prepared at an initial water content smallerthan the optimum water content (compacted dry of optimum) orlarger than the optimum water content (compacted wet of opti-mum). Reconstituted samples can be air-dried, heat-dried,freeze-dried or osmosis-dried. Different samples preparationmethods usually result in different soil microstructures. Forexample, soils compacted dry of optimum usually have a double-porosity microstructure, meaning that the pore size distributioncurve exhibits two or more peaks. In these soils, there are twotypes of pores: large inter-aggregates pores which are collapsibleupon wetting and small intra-aggregates pore which are morestable. On the other hand, soils air-dried from slurry usuallyexhibit a uni-modal pore size distribution, at least at low stresses.Nevertheless, as recently pointed out by Tarantino [133], theboundary between compacted and reconstituted soils is not alwaysclear and the microstructure of a soil can change with stress andhydraulic paths.

Unsaturated soils with a bi-modal pore size distribution (PSD)are usually collapsible. Wetting such a soil can collapse the inter-aggregates pores and results in a uni-modal pore size distributionwhen the soil becomes full saturated. In terms of constitutive mod-elling, a collapsible soil is characterised by a loading-collapse yield

Bi-modal: A

B: Uni-modal

s

p

C: Bi-modal?

LCA LCB

Fig. 12. Evolution of the yield surface for compacted soils according to Alonso et al.[2].

Slurry: B

s

p

C: Bi-modal?

LCCLCALCB

A

Fig. 13. Evolution of the yield surface for reconstituted soil according to Sheng et al.[108].


surface where the yield stress increases with increasing suction [2].This loading-collapse (LC) yield surface evolves with stress or suc-tion changes. As showed in Fig. 12, the soil at point A is unsatu-rated and has a bi-modal PSD and that at point B is saturatedand has a uni-modal PSD. The wetting path A ? B causes the LCcurve to evolve from LCA to LCB, as the inter-aggregates porescollapse and the soil volume decreases. However, drying theuni-modal soil at point B to point A and then compressing it topoint C, i.e. stress path B ? A ? C, should regenerate the bi-modalPSD, because the soil at point C becomes collapsible again accord-ing to the BBM [2]. Some experimental data also support such adevelopment of soil collapsibility [129]. In other words, the poresize distribution can evolve with stress and hydraulic paths. Mostconstitutive models in the literature are based on data for com-pacted soils and hence predict an evolution of the loading collapseyield surface as shown in Fig. 12.

Fig. 14. Oedometer curves for air-dry silt soaked

The question is now about the reconstituted soils. A soil recon-stituted from slurry is characterised by a uni-modal PSD, wetting itunder constant stress usually does not cause volume collapse.However, can such a soil become collapsible if it is air-dried andcompressed to high stresses? According to the SFG model [108],the yield surface for a slurry soil can evolve into a loading-collapsesurface if the soil is dried and compressed to sufficiently high stres-ses (point C in Fig. 13), which means that a uni-modal reconsti-tuted soil can evolve into a bi-modal collapsible soil (stress pathB ? A ? C). Is there experimental evidence for such an evolution?In other words, can drying and compressing of a reconstituted soillead to aggregation or cluster formation so that the soil becomescollapsible? Unfortunately there is very few data on reconstitutedsoils in the literature. Nevertheless, the classic reference by Jen-nings and Burland [48] seems to support such an evolution.Fig. 14 is a replot from Jennings and Burland [48] for an air-dry silt.

at various constant applied pressures [48].

Fig. 15. Isotropic compression curves for a reconstituted silty clay at varioussuctions [20].


It is clear that the air-dry soil can become collapsible if it is com-pressed to sufficiently high stresses. Another set of data reportedby Cunningham et al. [20] for a reconstituted silty clay also seemto indicate that compressing a soil at sufficiently high suctioncan eventually lead to a collapsible soil (Fig. 15). The limited datain the literature seem to support the evolution of LC surfaces assuggested in the SFG model (Fig. 13). However, experimental dataon reconstituted soils are generally too few to be conclusive.

In summary, the microstructure and particularly the pore sizedistribution of a soil is usually reflected in the yield surface andvolume change equation of the constitutive model for the soil. Abi-modal pore size distribution can evolve to a uni-modal pore sizedistribution under proper stress and hydraulic paths, and viceversa.

Table 1Shear strength equations and parameters used for compacted kaolin clay.

Equations tan ub/ tan u0 Parameters

1. Öberg and Sällfors [84] Sr SWCC2. Fredlund et al. [27] (Sr)j j = 1.3, hs = 0.659, SWCC3. Vanapalli et al. [140] h�hr

hs�hr

� �hs = 0.659, SWCC

4. Toll and Ong [137] Sr�Sr2Sr1�Sr2

� �k k = 1.2, SWCC

5. Alonso et al. [2] a a = 0.46. Sun et al. [122] a

sþa a = 110 kPa7. Khalili and Khabbaz [54] (sae/s)r sae = 60 kPa, r = 0.558. Sheng et al. [108] ssa

s þssas

� �ln s

ssa

� �ssa = 25 kPa

5. Shear strength

The change of shear strength with suction or saturation is one ofthe main reasons behind rainfall-induced landslides. It is related tothe volume change equation [110]. However, this relationship hasbeen overlooked in most existing models for unsaturated soils. Ifthe slope of the critical state line is assumed to be independentof suction, such as supported by experimental data of Ng andChiu [79], Thu et al. [135] and Nuth and Laloui [83], the shearstrength–suction relationship can be derived uniquely from thevolume change equation. If the slope of the critical state linedepends on suction, as supported by data of Toll [136], Toll andOng [137] and Merchán et al. [74], two equations are needed todefine the shear strength–suction relationship, namely the volumechange equation, and the M(s) function, with M being the slope ofthe critical state line in the deviator – mean stress space.

Bishop and Blight [8] first proposed an effective stress definitionto interpret the shear strength of unsaturated soils:

s ¼ c0 þ r0n tan /0 ¼ c0 þ ð�rn þ vsÞ tan /0 ¼ �c þ �rn tan /0 ð26Þ

where s is the shear strength, c0 is the effective cohesion for satu-rated states and is usually assumed to be zero, r0n and �rn are respec-tively the effective and net normal stress on the failure plane, u0 isthe effective friction angle of the soil, v is the well-known Bishop’seffective stress parameter, and �c is the apparent cohesion which in-cludes the friction term due to suction, i.e. �c ¼ c0 þ vs tan /0.

Fredlund et al. [25] proposed the following relationship whichconveniently separates the shear strength due to stress from thatdue to suction:

s ¼ ½c0 þ ðrn � uaÞ tan /0� þ ½ðua � uwÞ tan /b�¼ �c þ ðrn � uaÞ tan /0 ð27Þ

where s is the shear strength, c0 the effective cohesion for saturatedstates and is usually assumed to be zero, rn the normal stress on thefailure plane, u0 the effective friction angle of the soil, and ub thefrictional angle due to suction. Obviously, if ub is set to u0 in Eq.(27), the Coulomb friction criterion in terms of effective stress forsaturated soils is recovered.

A common conception is that the shear strength of an unsatu-rated soil can be sufficiently defined by a single effective stress[3,105,146]. However, this conception is only true when the slopeof the critical state line or the friction angle of the soil does notchange with suction. When the friction angle of the soil changeswith suction, the shear strength can no longer be defined by a sin-gle effective stress. This is clear by comparing equation (26) with(27):

s ¼ ½c0 þ �rn tan /0ðsÞ� þ ½s tan /b�

¼ c0 þ �rn þ stan /b

tan/0ðsÞ

!tan /0ðsÞ ¼ c0 þ r0n tan /0ðsÞ ð28Þ

where the Bishop effective stress parameter (v) is set to tan ub/tan u0. It is clear that the variable (s) cannot be eliminated no mat-ter how the effective stress is defined.

The shear strength of an unsaturated soil is fully defined if theapparent cohesion (�c) or the friction angle due to suction (ub),and the friction angle due to stress u0 in Eq. (27) are known. Anumber of alternative equations have been used in the literatureto define �c or ub (e.g. [27,54,75,84,137,140]). Most of these equa-tions are empirically based and are defined independently fromthe volume change equation, which, if incorporated into a constitu-tive model, may lead to inconsistency with the yield surface as dis-cussed in the Section above. Rojas [94,95] presented an analyticalexpression for Bishop’s parameter v in terms of the saturated frac-tion and Sr of the unsaturated fraction of the soil and used it tointerpret shear strength. One issue with this approach is that vcannot be easily determined either experimentally or theoretically.A similar approach was proposed by Pereira and Alonso [90] wherev is expressed in terms of the effective degree of saturation thatdepends on the micro- and macro-saturation.

Sheng et al. [115] recently compared various shear strengthequations for unsaturated soils against a large number of data sets.The equations include those empirically based and those embed-ded in constitutive models. The equations studied by Sheng et al.[115] are listed in Table 1. Fig. 16 presents one example of suchcomparisons. The parameters used in the shear strength equations

600

500

400

300

200 0 100 200 300 4000 100 200 300 400

600

500

400

300

200

Dev

iato

r Stre

ss, k

Pa

Suction, kPa Suction, kPa

Predictions Test data (after Thu et al. 2007b)

1

Confining pressure: 100 kPa

7

6

5

32

4

8

Test data (after Thu et al. 2007b)

Confining pressure: 200 kPa

Predictions

7 8

5

1

24

3

6

Fig. 16. Predictions of triaxial test data on compacted kaolin clay by various shear strength equations (data by [135]). Note that the prediction by Khalili and Khabbaz [54], i.e.No. [7], can be improved significantly by using a lower air entry value.


are given in Table 1. The comparative study by Sheng et al. [115]reveals that:

1. If the friction angle of the soil is assumed to be independent ofsuction, all shear strength equations in the literature can be for-mulated in form of either Eq. (26) or Eq. (27). In this case, thereis little difference in formulating shear strength in one singlestress variable or in two independent stress variables. The realchallenge is to find a single effective stress when the frictionangle depends on suction.

2. The performance of the shear strength equations in predictingexperimental data depends on the careful determination ofmaterial parameters and also on the specific data set. A shearstrength equation may predict one data set better than otherdata sets.

3. The shear strength equations that incorporate the soil–watercharacteristic curve (SWCC) generally require more parameters.This group of equations seem to provide reasonable predictionof shear strength for unsaturated soils (e.g. [27,140,137]). How-ever, some equations are sensitive to the residual suction[140,137], which can be difficult to determine accurately fromthe SWCC.

4. Simpler shear strength equations that are embedded in consti-tutive models appear to provide reasonable predictions of shearstrength [54,108,122]. These equations contain typically one ortwo parameters. However, these equations usually do not pre-dict any peak value of the shear strength attained at an interme-diate suction level.

6. Water retention behaviour and hydro-mechanical coupling

The issue of interaction between the mechanical and hydraulicbehaviour was perhaps first raised by Wheeler [145] and then byDangla et al. [22]. The first complete model that deals with coupledhydro-mechanical behaviour of unsaturated soils was perhaps dueto Vaunat et al. [143]. A number of coupled models soon followed(e.g. [114,149]). With respect to hydraulic behaviour of unsatu-rated soils, many models [28,139] take advantage of the fact thatthe influence of suction on degree of saturation is more significantthan the influence of deformation. The dependency of degree ofsaturation on suction is described by a soil–water characteristiccurve (also called soil–water retention curve, SWRC). Only until

recently, the effects of deformation on SWCCs have been consid-ered (e.g. [31,76,124,149,160]).

As pointed out by Wheeler et al. [149], the mechanical behav-iour of an unsaturated soil depends on degree of saturation evenif the suction, net stress and specific volume are kept the samefor the soil. Separate treatment of mechanical and hydraulic com-ponents in modelling unsaturated soil behaviour has certain limi-tations in reproducing experimental observation. It would bedifficult to consider the saturation dependency in a mechanicalmodel that is independent of the hydraulic behaviour. Similarly,a hydraulic model that is independent of mechanical behaviourcannot easily take into account the effects of soil density on theSWCC. Experimental data generally demonstrate the followingpoints:

1. A SWCC obtained under a higher net mean stress tends to shifttowards the higher suction [31,62,73,81]. This means that theincremental relationship between degree of saturation (Sr)and suction (s) depends on net mean stress (�p) or soil density.

2. When the suction is kept constant, isotropic loading or unload-ing can also change the degree of saturation of an unsaturatedsoil [149]. This implies that the degree of saturation is relatedto stress or soil density when the suction is kept constant.

One of the early models that fully couple the hydraulic andmechanical components of unsaturated soil behaviour is that byWheeler et al. [149]. Models that appeared before or soon afterWheeler et al. [149] tend to accentuate the influences of thehydraulic component on the mechanical component, not vice versa(e.g. [83,114,143]). The interaction between the mechanical andhydraulic components in the model by Wheeler et al. [149] wasrealised through the use of the average soil skeleton stress (effec-tive stress), the modified suction and the coupling between theloading-collapse and suction-increase and suction-decrease sur-faces. The average soil skeleton stress ðr�ijÞ is an amalgam of stress,suction and degree of saturation. The modified suction (s�) is acombination of suction and porosity. Therefore, the influence ofhydraulic behaviour on the stress–strain relationship is consideredvia the definition of the average stress. The influence of porosity onthe saturation–suction relationship is considered via the definitionof the modified suction. The model by Wheeler et al. [149] is one ofthe few models that are qualitatively tenable in terms of couplingmechanical behaviour with hydraulic behaviour for unsaturated


soils. However, the use of the modified suction and the soil skele-ton stress, which is one of the advantages that makes the modelrigorously consistent in thermodynamics, can become a disadvan-tage as well, particularly in terms of quantitative prediction andthe application of the model. For example, one of the difficultiesin using this model is to quantify the synchronised movement be-tween the loading-collapse (LC) surface and the suction-increase(SI) and suction-decrease (SD) surfaces. This synchronicity cannoteasily be calibrated by laboratory experiments [93] or definedtheoretically.

In more recent models, the influences of mechanical propertieson the hydraulic behaviour are usually modelled via the depen-dency of the SWCC on soil volume [31,132], soil density [71,124],or volumetric strain [82]. Gallipoli et al. [31] suggested includinga function of specific volume (v) in the SWCC equation of vanGenuchten [139]. Tarantino [132] showed there is a unique rela-tionship between the water ratio (product of Sr and e) and the mat-ric suction and used this relationship to modify van Genuchten’sequation. The modified van Genuchten’s equation takes a similarform as Gallipoli et al. [31]. It is also common to express the SWCCequation in incremental forms. For example, Sun et al. [124] pro-posed a hydraulic model in the following form:

dSr ¼ ksede� kssds=s ð29Þ

where kss is the slope of main drying or wetting curve, and kse theslope of degree of saturation versus void ratio curve under constantsuction. In theory, kss in the equation above can only be determinedfrom constant-volume tests (de = 0). Mašín [71] used a similarequation as (29). In his model both air entry value (sae) and theslope of main drying curve (kss) vary with void ratio. Nuth andLaloui [82] provided an alternative approach of modelling SWCCfor a deforming soil. They assume there is an intrinsic SWCC for anon-deforming soil and deformation of the soil can shift this intrin-sic SWCC along the suction axis. The shift is governed by an air en-try value that depends on the volumetric strain.

The models by Sun et al. [124], Nuth and Laloui [82], Mašín [71]and many others (e.g. [53,83,108,114,128,160]) essentially alladopt a water retention equation in the following form:

dSr ¼ ð. . .Þdsþ ð. . .Þdev ð30Þ

This equation is not wrong, but the embedded Sr–s relationship isfor constant volume (dev = 0). Therefore, it does not recover the con-ventional SWCC equations, which are obtained under constantstress. The volume change along a conventional SWCC can be signif-icant, particularly at low suctions. For expansive soils, it is also com-mon to study the water retention behaviour under constant volume[65,96,98]. On the other hand, if one specific Sr–s relation is valid for

A

B

LC yield surface

SI surface

SD surface

s

p

C D

A

Main d

Main wetti

s

C D

(a) stress path in net stress-suction space

(b) inconsistent

A

Fig. 17. Qualitative analysis of isotropic co

constant stress, the Sr–s relation for constant volume would inevita-bly involve soil compressibility and hence stress (suction) level andstress (suction) history. Therefore, it is unlikely that the same Sr–srelation can be used to calibrate water retention data both for con-stant stress and constant volume. Furthermore, neglecting the vol-ume change along SWCC can lead to inconsistent prediction of thedegree of saturation during undrained compression, an issue raisedby Zhang and Lytton [154].

Sheng and Zhou [116] proposed a new method for couplinghydraulic with mechanical behaviour. This new method is basedon the fact that SWCCs are obtained under constant stress. InSheng and Zhou [116], the volume change behaviour and the waterretention behaviour under isotropic stress states are representedby the following incremental equations, respectively:

dev ¼ Ad�pþ Bds ð31Þ

dSr ¼ Edsþ Sr

nð1� SrÞfAd�p

¼ E� BSr

nð1� SrÞf

� �ds� Sr

eð1� SrÞfde ð32Þ

where parameters A and B are related to the specific volume changeequation as discussed in Section 3, parameter E is the gradient ofthe conventional SWCC, e is the void ratio, n is the porosity, and fis a fitting parameter. If the SFG model is used to describe the vol-ume change, parameters A and B would then take the followingform respectively:

A ¼ kvp

�pþ f ðsÞ ; B ¼ kvs

�pþ f ðsÞ ð33Þ

The Sr–s relationship in Eq. (32) is defined for constant stress(d�p ¼ 0) and hence parameter E refers to the gradient of the conven-tional SWCC:

E ¼d SSWCC

r ðsÞ� �

dsð34Þ

where SSWCCr represents the conventional SWCC equation. The void

ratio in Eq. (32) refers to the value at the current stress and suction.

It is clear from Eq. (32) that the Sr–s relationship for constantvolume (de = 0) is more complex than the conventional SWCCequation (dSr ¼ Eds). Eq. (32) was proposed based on experimentalobservation as well as the intrinsic phase relationship for un-drained condition:

dSr ¼Sr

ndev; dw ¼ 0 ð35Þ

A

B

Main drying curve

Main wetting curve

s

Sr

C

D B

rying curve

ng curve

Sr

Scanning curve

change of Sr (c) consistent change of Sr

mpression under undrained condition.

0.30

0.20

0.10

0.00

S r 7-10-U

Calibration: 5-10-K

7-10-V

7-10-P

7-10-W 5-10-L

5-10-I

Predictions

ζ = 0.1

1.2 1.4 1.6 1.8 1+e

Fig. 18. Validation of Eq. (37) using suction-controlled oedometer compressiondata of a compacted soil (data after [51]).

0.6

0.7

0.8

0.9

1

1.1

0.1 1 10s (kPa)

100 1000 10000

e0 =0.444

e0 =0.474 e0 =0.514

e0 =0.517

Predictions

S r

Fig. 19. Predicted and measured SWCCs at various initial void ratios of a compactedtill (data after [141]).


where w is the gravimetric water content. Eq. (35) actually imposesa constraint on suction change under undrained compression. Thisconstraint is obtained by substituting Eq. (35) into Eq. (32):

ðSr � Srð1� SrÞfÞAd�p ¼ ðnE� BÞds; dw ¼ 0 ð36Þ

Zhang and Lytton [154] recently noted that some models fall shortin predicting undrained behaviour of unsaturated soils. The specificissue raised is illustrated in Fig. 17. Assume the initial state of a soilis inside the elastic zone, but on the main wetting curve, i.e. point Ain Fig. 17a. Compressing the soil under undrained condition willlead to some suction decrease [127,131], say to point B. Assume Bis still inside the elastic zone. Unloading from B to A will recoverthe initial volume of the soil and hence the initial degree of satura-tion should be recovered as well. However, if the volume changealong the SWCC is neglected, the change of Sr along path ACB wouldfollow the main wetting curve and is hence ‘elastoplastic’, whereasthe change of Sr along path BDA would follow the scanning curveand is hence ‘elastic’, leading to inconsistent change of Sr

(Fig. 17b). It would then seem unlikely that a model where the irre-versible volume change is not synchronised with the irreversiblesaturation change could lead to a consistent prediction of saturationchange over the closed path ABA [155]. This inconsistency is actu-ally due to the assumption that the main wetting curve is definedfor constant volume.

If Eq. (32) is used, the inconsistence in Fig. 17b will be avoided.Because of Eq. (35), the model will predict no change of Sr as longas ev = 0, irrespective of evolution of the suction-decrease surface.Eq. (35) is satisfied as long as the suction changes according to(36). It is easy to understand that the loading path ACB is not onthe initial main wetting curve and the unloading path BDA is noton the scanning curve, because the mean stress is not constantalong those stress paths. The main wetting curve at point A alsoshifts to that at point B, as the mean stress changes (Fig. 17c).The synchronised evolution of the LC, SI and SD surfaces (as in[149] is not necessary for the consistent prediction in Fig. 17c. In-deed, the suction path can be ‘elastoplastic’ even though the stresspath is elastic.

The following equation can be derived from Eq. (32):

@Sr

@e¼ � Srð1� SrÞf

e; ds ¼ 0 ð37Þ

The void ratio (e) in Eq. (37) refers to the initial void ratio at the cur-rent stress and its variation is purely due to stress change. It canalso be interpreted as the initial void ratio at the start of the SWCCtests. Eq. (37) shows that the SWCC for a soil shifts with its initialvoid ratio. This is similar to the approach by Gallipoli et al. [31]where the van Genuchten equation was modified to include the ini-tial void ratio. Indeed, their SWCC equation can be re-written as:

@Sr

@e¼ �mnw

Srð1� S1=mr Þ

eð38Þ

where m and n are two fitting parameters in the original vanGenuchten equation, and w is another parameter introduced byGallipoli et al. [31]. If the product (mnw) was set to 1, Eq. (38) wouldbe equivalent to (37). Sheng and Zhou [116] showed that the intrin-sic phase relationship requires:

1� Sr

eP

@Sr

@eP�Sr

eð39Þ

The above constraint is satisfied if (mnw) = 1 in Eq. (38). It is alsointeresting to note that a value of 1.1 for (mnw was used in allthe numerical examples in Gallipoli et al. [31].

Eq. (37) can be integrated either analytically for certain f valuesor numerically in more general cases. Because Eq. (37) is in anincremental form, integration of the equation requires one specific

SWCC equation that corresponds to a reference initial void ratio. Inother words, the conventional SWCC equation is only used for thereference initial void ratio and the new SWCC for a new initial voidratio is obtained by integration of (37).

The model by Sheng and Zhou [116] is validated against a vari-ety of data sets. In Fig. 18, the results from suction-controlledoedometer compression tests by Jotisankasa [51] are used to vali-date Eq. (37). Since the suction remains constant for each curve inFig. 18, no SWCC equation is required here and the only parameterrequired is f. Eq. (37) can directly be used to predict the variationof degree of saturation due the variation of stress-induced volumechange. In this case, the void ratio (e) in Eq. (37) is the current voidratio and it changes due to stress variation in the oedometer tests.Fig. 18 clearly shows that Eq. (37) predicts very well the relation-ship between the degree of saturation and the void ratio whenthe suction is kept constant. Indeed, the experimental data are al-most exactly on the predicted Sr–e curves. In Fig. 19, the data byVanapalli et al. [141] on a compacted till were used. The SWCCfor e0 = 0.517 was fitted by the van Genuchten equation, whilethe other SWCCs are predicted by Eq. (37) with f = 0.03. Fig. 19shows that both the slope and the air entry value of the SWCCschange with the initial void ratio. In the two cases studied, themodel by Sheng and Zhou [116] seems to be able to capture the ef-fect of initial void ratio on the soil water retention behaviour. One

45o p

s

Elastic zone

p′

s

Elastic zone

unsaturated

saturated p

saturated

Elastic trial stress

Elastic zone

unsaturated

s

45o

(a) net stress (Alonso et al.,1990)

(b) effective stress (Sheng et al., 2004)

(c) net stress (Sheng et al., 2008a)

unsaturated

saturated

LC surface

Fig. 20. Non-convexity of yield surfaces in unsaturated soil models in the suction–stress space.

f

0 α1

1/21/41/8

Fig. 21. Bracketing the roots for nonlinear function according to Pedroso et al.[87,88].


advantage of this approach is that Eq. (37) can be used with anyexisting SWCC equations, including those for uni-modal pore sizedistribution (e.g. [28,139]) and those for bi-modal pore size distri-bution (e.g. [11,40]), as well as those advanced models that adopthysteretic scanning loops within the main drying-wetting loop[64,87–89].

7. Finite element implementation

One of the ultimate goals of constitutive modelling is to imple-ment the model in a numerical method to solve boundary valueproblems. A constitutive model can generally be formulated inthe following incremental form [66,110,114]:

d�rdh

� �¼ Dep Wep

T H

� �deds

� �ð40Þ

where �r is the stress vector, e is the vector of soil skeleton strain, h isthe volumetric water content, Dep, Wep, T and H are constitutivematrices [110].

In the displacement finite element method, the nodal displace-ments and pore pressures are first solved from the equilibrium andcontinuity equations, based on the current stress states and thecurrent volumetric water content. The strain and suction incre-ments are then derived from the displacements and pore pres-sures. For given strain and suction increments, the current stressvector, the volumetric water content and the internal variablesmust be updated according to Eq. (40). This update is generally car-ried out by numerical integration.

One of the main challenges in integrating Eq. (40) arises fromthe non-convexity of the yield surface around the transition be-tween saturated and unsaturated states [106,113,147]. The non-convexity seems to persist irrespective of the stress variables andis demonstrated in Fig. 20 [111].

Both implicit and explicit schemes have been used to integrateunsaturated soils models. In implicit schemes, all gradients andfunctions are estimated at advanced unknown stress states andthe solution is achieved by iteration. This group of stress integra-tion schemes include Vaunat et al. [142], Borja [10], Hoyos andArduino [46], Zhang and Zhou [153], and Tamagnini and De Genna-ro [130]. They usually do not deal with the non-convex problem byavoiding the transition between saturated and unsaturated states.Borja [10] was perhaps the only one in this group who noticed theproblem. He suggested keeping the step size sufficiently small toavoid overshooting the plastic zone by the elastic trial stress(Fig. 20). Otherwise, there is currently no general method used totackle the non-convexity problem in implicit schemes.

On the other hand, explicit schemes estimate the gradients andfunctions at the current known stress states and proceed in anincremental fashion. These methods are theoretically more

appropriate for non-convex models [87,88]. González and Gens[42] compared both an implicit and an explicit scheme for integrat-ing the BBM and found that the latter is more robust and efficient.However, explicit schemes usually require to determine the inter-section between the current yield surface and the elastic trialstress path and some substepping methods to control the integra-tion error [102,110,111,112,113,119,120].

A key issue in integrating the incremental stress–strain rela-tionships using an explicit scheme is thus to find the intersectionbetween the elastic trial stress path and the current yield surface.The most complex situation occurs when the yield surface iscrossed more than once, such as shown in Fig. 20. However, itis not possible to know a priori how many times the yield surfaceis crossed. Therefore, for non-convex yield surfaces, the keytask is to find the very first intersection for any possible stressand hydraulic path.

Finding the intersection between the elastic trial stress incre-ment and the current yield surface can be cast as a problem of find-ing the multiple roots of a nonlinear equation:

f ðaÞ ¼ f ð�ra; sa; zaÞ ¼ 0 ð41Þ

where 0 6 a 6 1; f ð�r; s; zÞ is the yield function, z is a set of internalvariables, and subscript a indicates the quantity is estimated atstrain increment aDe and suction increment aDs. Pedroso et al.[87,88] proposed a novel method to bracket the roots (a). Themethod is illustrated in Fig. 21. For a given increment (a = 1), thenumber of roots of f(a) is first computed. If there is more thanone root, the increment is divided into two equal sub-increments.The number of roots of each sub-increment is then computed. Ifthe first sub-increment contains more than one root, it is furtherdivided into two sub-increments. This process is repeated untilthe first sub-increment contains at most one root (Fig. 21). Oncethe roots are bracketed, the solution of the first root can be foundby using numerical methods such as the Pegasus method [118].

Sheng et al. [111] applied the root bracketing method by Pedr-oso et al. [87,88] to integrate the SFG model and found that themethod can indeed provide an accurate solution of the intersection


problem. However, this method is found to be computationallyexpensive. The formula used to find the number of roots of a non-linear function requires both the first and second orders of gradi-ents of the function and requires numerical integration.Furthermore, the root-finding procedure must be applied for allsuction increments near the non-convexity. Further research inthis direction is clearly required.

8. Conclusions

Some conclusions can be drawn from this review of constitutivemodelling of unsaturated soils:

1. Partial saturation is only a state of soil. Constitutive models forsoils should represent the soil behaviour over entire ranges ofpossible stress and pore pressure values. This requires a consis-tent and coherent merge of fundamental soil mechanics princi-ples for both saturated and unsaturated states.

2. The volume change behaviour is one of the most fundamentalproperties of soils. For soils in unsaturated states, the volumechange equation also underpins the yield stress–suction andshear strength–suction relationships. It also affects the soilwater retention behaviour. Indeed, one of the most essentialdifferences amongst various existing models is the volumechange equation. Other differences are often the consequencesof the volume change equation.

3. Three groups of volumetric models are compared and it isshown that each has advantages and disadvantages. There iscertainly room for improvement of all these models. One obser-vation here is that it seems difficult to describe the volumechange of unsaturated soils in terms of one single stressvariable.

4. The methods used to define the loading-collapse, suction-increase and apparent tensile strength surfaces should beconsistent with the volume change equation. In addition, thesesurfaces and the shear strength function are all related to eachother and hence should be defined consistently with each other.

5. It seems possible to model reconstituted soils and compactedsoils within the same theoretical framework. The pore size dis-tribution evolves with mechanical and hydraulic loading and isreflected by the loading collapse yield surface and the volumechange equation.

6. If the friction angle of the soil is assumed to be independent ofsuction, all shear strength equations in the literature can be for-mulated either in terms of one single stress variable or in termsof two independent stress variables. The real challenge is to finda single effective stress when the friction angle depends onsuction.

7. The performance of the shear strength equations in predictingexperimental data depends on the careful determination ofmaterial parameters and also on the specific data set. A shearstrength equation may predict one data set better than otherdata sets.

8. When coupling the hydraulic component with the mechanicalcomponent in a constitutive model, it is recommended to takeinto account the volume change along soil–water characteristiccurves. Neglecting this volume change can lead to inconsistentprediction of volume and saturation changes.

9. Unsaturated soil models are characterised by non-convex yieldsurfaces at the transition between saturated and unsaturatedstates. This non-convexity, if handled rigorously, can signifi-cantly complicate the implementation of these models intofinite element codes. In this case, an explicit stress integrationscheme incorporating an efficient root finding algorithm ispreferred.

Acknowledgements

The author would like thank the editor of Computers and Geo-technics for the invitation of this review paper. This paper is basedon the General Report presented at the 5th International Conferenceon Unsaturated Soils held at Barcelona during 6–8 September 2010[105]. A number of people provided valuable comments and con-tributions to the General Report: Y.J. Cui, D.G. Fredlund, D. Gallipol-i, A. Gens, S.L. Houston, J. Kodikara, J.M. Pereira, W.T. Solowski, D.A.Sun, C. Yang, X. Zhang and A.N. Zhou. The author is grateful to all ofthem. The General Report has since been substantially revised toproduce this paper, partly inspired by questions and informal dis-cussions during and after the Barcelona conference. Finally thefinancial support from the Australian Research Council is greatlyappreciated.

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Documents

Computers and Geotechnics - University of Newcastle · and Jommi [97] and D’Onza et al. [24] for anisotropy; Modaressi and Modaressi [77] and Cui et al. [18] for non-isothermal