Soil stability and ßow slides in unsaturated shallow … Prisco... · Soil stability and ßow...

Buscarnera, G. & di Prisco, C. Geotechnique [http://dx.doi.org/10.1680/geot.11.P.097]

1

Soil stability and flow slides in unsaturated shallow slopes: can saturationevents trigger liquefaction processes?

G. BUSCARNERA� and C. DI PRISCO†

This paper illustrates an application of the theory of material stability to the analysis of unsaturatedslopes. The main goal is to contribute to the understanding of rainfall-induced flow slides. For thispurpose, a coupled hydromechanical constitutive model is combined with a simplified approach forthe analysis of infinite slopes. Simple shear-test simulations are used to evaluate triggering perturba-tions and investigate the role of both initial suction and stress anisotropy in the activation of slopefailures. The numerical simulations clearly show that different mechanisms of activation can beoriginated. The onset of instability is detected by introducing appropriate stability indices for distinctmodes of failure: localised shear failure, static liquefaction and wetting-induced collapse. Criticalintervals of slope inclinations are identified, cautioning that the predicted failure mode may changedramatically depending on initial conditions, slope angle and material properties. The numericalsimulations demonstrate that, in particular circumstances, saturation of the pore space can be theunexpected result of a volumetric instability. According to this interpretation, a rainfall-induced flowslide can originate from a complex chain process consisting of a sudden volume collapse, uncontrolledsaturation of the pores and, eventually, catastrophic liquefaction of the deposit.

KEYWORDS: constitutive relations; landslides; liquefaction; partial saturation; suction; theoretical analysis

INTRODUCTIONIn many parts of the world, geohazards pose serious threatsto territory, economy and human lives. During recent dec-ades, catastrophic events have been exacerbated by unpre-dicted climate changes and uncontrolled human activities(Cascini, 2005). The environment tends to be exposed tophenomena never experienced before, which now representsystematic causes of massive economic loss. Within thiscontext, rapid landslides induced by rainfall represent acritical issue. These catastrophic events are characterised byrapid and unexpected activation, and are capable of mobilis-ing huge volumes of material over large areas (Chu et al.,2003; Olivares & Picarelli, 2003; Picarelli et al., 2008).

The compelling need to capture the physical causes ofsuch dramatic landslides requires a deep understanding ofthe phenomena involved, and advanced modelling strategies.This paper focuses on the study of landslides triggered byrainfall events, with the aim of investigating the mechanicsof these processes and modelling their activation. Particularemphasis is given to the study of those landslides in whichthe soil suffers a phase transition from solid to fluid (herereferred to as ‘flow slides’). Such transition is usuallyattributed to a liquefaction process (Castro, 1969; Lade,1992), which is schematically illustrated in Fig. 1. Depend-ing on the shearing scenario (either undrained or drained),different failure modes can take place, given that in aliquefiable deposit the shear perturbations leading to lique-faction (˜�liq) are significantly lower than those associatedwith drained failures (˜�sf ). The main engineering implica-tion of such a variety of instability modes is the existence of

Manuscript received 29 July 2011; revised manuscript accepted 7December 2012.Discussion on this paper is welcomed by the editor.� Department of Civil and Environmental Engineering, NorthwesternUniversity, Evanston, USA.† Department of Structural Engineering, Politecnico di Milano,Milan, Italy.

Soil

Bedrock

z

Shear perturbation ( )Δτ

σ �n0

τ0

α

(a)

(b)

Δτsf

Δτliq

She

ar s

tres

s,τ

Normal effective stress, σ �n

In situ stress

Shear failure locus

Fig. 1. (a) Schematic representation of a shear perturbation ˜�acting over a submerged infinite slope (�9n is the in situ normaleffective stress); (b) possible failure modes: difference in externalperturbations needed to activate either drained shear failure(˜�sf ) or static liquefaction (˜�liq)

a multiplicity of safety factors. These considerations inspiredthe development of a wide spectrum of theories aimed atdifferentiating liquefaction from shear failure and evaluatingthe risk of flow slides in subaqueous sandy slopes (Poulos etal., 1985; Sladen et al., 1985; di Prisco et al., 1995).

In the reminder of the paper this logic will be exported tothe case of sub-aerial slopes, for which a comprehensiveframework of interpretation is not yet available. The purposeis to provide a consistent geomechanical explanation offailure processes in unsaturated deposits by answering threemajor questions.

(a) What is the amount of suction removal at which a slidecan take place?

(b) How and when can slope failures evolve into a flow-likemass movement?

(c) Are fully saturated conditions necessary to induceliquefaction, or can such collapses be initiated by theprior wetting process?

To address such problems, a novel methodology has beendeveloped, which is based on three cornerstones:

(a) the extension of the concepts of material stability tounsaturated soils (Buscarnera & Nova, 2011; Buscarnera& di Prisco, 2012)

(b) their application to the simplified scheme of infinite slope(di Prisco et al., 1995; Buscarnera & di Prisco, 2011a,2011b; Buscarnera & Whittle, 2012)

(c) the use of a coupled hydromechanical constitutive modelfor unsaturated soils (Buscarnera & Nova, 2009).

The paper is thus aimed at applying the theory alreadydiscussed in Buscarnera & di Prisco (2011b) by employing asuitable constitutive relationship. The main goal of theinvestigation is to elucidate the mechanical processes in-volved in the triggering of flow slides in partially saturatedsoil slopes.

THEORETICAL BACKGROUNDSecond-order work and controllability for unsaturated soils

A generally accepted approach for identifying unstableconditions in solids is Hill’s criterion (Hill, 1958), accordingto which a sufficient condition for stability is the positivedefiniteness of the second-order work, d2W. This criterionprovides a physical interpretation for instability, given thatnegative values for d2W can correspond to a spontaneousburst in kinetic energy (Sibille et al., 2007), and can be usedfor studying the initiation of slope instabilities (Lignon etal., 2009). In unsaturated contexts, a critical task is toincorporate the mechanical implications of saturation pro-cesses. In fact, changes in the degree of saturation implyadditional energy contributions (Houlsby, 1997; Gray et al.,2010), and require the adaptation of second-order workmeasures. A strategy for this extension has been recentlysuggested by Buscarnera & di Prisco (2012), who showedthat under unsaturated conditions d2W becomes

d2W ¼ 12

_�ij � Sr _uw�ij � 1� Srð Þ _ua�ij

� �_�ij � 1

2n _ua � _uwð Þ _Sr

¼ 12

_��ij _�ij � 12_s� _Sr

(1)

where �ij is the total stress tensor, �ij is the strain tensor, �ij

is Kronecker’s delta, _uw and _ua are the pore water and poreair pressure rates respectively, Sr is the degree of saturation,and n is the porosity. For Sr ¼ 1, the above expressionconverges to the usual definition of second-order work forsaturated media (i.e. d2W ¼ 1

2_� 9ij _�ij, � 9ij ¼ � ij � uw�ij being

the effective stress tensor). By rearranging equation (1) it is

also possible to identify incremental stress variables forsecond-order work analyses, as for instance

_��ij ¼ _�ij � Sr _uw�ij � 1� Srð Þ _ua�ij

and

_s� ¼ n_s ¼ n _ua � _uwð ÞThe extended expression for d2W enables instability condi-

tions for unsaturated geomaterials to be identified and linkedto the mathematical concept of controllability (Nova, 1994;Imposimato & Nova, 1998; Buscarnera & Nova, 2011;Buscarnera et al., 2011). In order to describe this concept,consider a set of incremental hydromechanical constitutiverelations linking the control variables _� (i.e. the disturbanceapplied to the material) and the response variables _ł (i.e.the outcome of the response of the material), as

_� ¼ X _ł (2)

where X is the control matrix. If the hydromechanicalvariables in equation (2) are selected in accordance withequation (1), the loss of uniqueness and/or existence of theincremental response (i.e. det X ¼ 0) corresponds to a van-ishing second-order work (i.e. it violates Hill’s criterion).

The concept of controllability

(a) provides a further insight into the physical meaning ofequation (1)

(b) defines the failure mode mathematically (through theeigenvectors of X)

(c) permits an intuitive definition of instability that encom-passes saturated (Imposimato & Nova, 1998) andunsaturated conditions (Buscarnera & Nova, 2011).

As will be expounded later, the most notable feature of thistheory is the ability to cope with ‘latent instabilities’, that is,potential collapses that are contingent on specific boundary/control conditions (Nova, 1994; di Prisco et al., 1995;Buscarnera & Whittle, 2013).

Using second-order work principles in unsaturated infiniteslopes

The concepts of second-order work and controllability canbe used to elucidate the hydromechanical properties that canplay a role in the initiation of uncontrolled deformationprocesses. For this purpose, consider a reference systemassociated with an unsaturated infinite slope of a giveninclination (Fig. 2). By starting from the three-dimensionalstress–strain response of a material point (see the Appendixfor such a representation), it is possible to represent the

Soil

Bedrock

α

�

η

� (out-of-plane coordinate)

Fig. 2. Reference system for unsaturated infinite slopes

2 BUSCARNERA AND DI PRISCO

mechanical response of a point within the deposit as asimple shear deformation mode, having

_� ¼_� ��_��

_s�

8><>:

9>=>; ¼

D11 D14 D17

D41 D44 D47

D71 D74 D77

24

35 _��

_ª�� _Sr

8><>:

9>=>; ¼ D _E (3)

The two vectors _� and _E are linked by a coupled constitu-tive operator (matrix D) and collect the hydromechanicalvariables associated with the incremental energy input on aninfinite slope. In particular, _�� is the incremental skeletonstress along the direction normal to the slope, and _�� is theshear stress increment along the slope inclination, while _��and _ª�� are their work-conjugate kinematic counterparts.The hydraulic variables associated with changes in saturationconditions (i.e. _s� and _Sr) are selected on the basis ofequation (1).

The loss of positive definiteness of d2W is governed bythe symmetry properties of D. By decomposing this matrixinto the sum of a symmetric part, Ds, and a skew-symmetricmatrix, Dsk, it can be shown that

d2W ¼ 12_�

T _E ¼ 12

_ETDs _E (4)

given that Hill’s criterion can be violated for the first timewhen det Ds ¼ 0 (Imposimato & Nova, 1998). It interestingto observe that, since det Ds < det D (Ostrowski & Taussky,1951), possible non-symmetries of D imply that Hill’s criter-ion can be violated before condition det D ¼ 0 is satisfied.As a result, stress-suction control conditions may not be themost critical combination of control parameters underpinningthe collapse of natural slopes, which thus suggests theexistence of unexpected failure modes. For simple shearconditions this concept is exemplified by the skew-sym-metric part, Dsk, which embodies the difference betweenmatrix D (whose singularity reflects suction-controlled fail-ure) and its symmetric part Ds (which reflects possibleviolations of Hill’s criterion)

Dsk ¼

0D14 � D41

2

D17 � D71

2D41 � D14

20

D47 � D74

2D71 � D17

2

D74 � D47

20

2666664

3777775 (5)

Alternative modes of failure are promoted by the differ-ence between terms [D14, D41] (reflecting non-associativityof the mechanical response) or between the pairs [D17, D71]and [D74, D47] (related to hydromechanical coupling). While[D14, D41] depend on the characteristics of the yield surfaceand the plastic potential, the other off-diagonal terms reflectdifferent behavioural properties: terms D17 and D47 repro-duce the inelastic effects of saturation paths (e.g. wetting-induced compaction), and terms D71 and D74 reproduceinstead the dependence of the retention curve on volumetricand shear strains respectively. At variance with the effect ofnon-associativity on material instabilities, which has beenwidely studied for several decades (Rudnicki & Rice, 1975;Bigoni & Hueckel, 1991; Lade, 1992; Nova, 1994), the roleof hydromechanical coupling still deserves special attention.Similar to the non-associativity of the plastic flow rule,hydraulic off-diagonal contributions must be assessed on thebasis of experimental evidence. In this work, the assump-tions for the hydromechanical contributions are motivated bythe geomechanical characterisation of some unsaturated soilsinvolved in recent flow slide events (Cascini & Sorbino,2004; Bilotta et al., 2005; Ferrari et al., 2012). These studiessuggest that, while suction effects can induce changes in thepreconsolidation stress of the collapsible deposits (terms D17

and D47), the effect of soil deformation on the retentioncurve is often negligible at shallow depths. It is worth notingthat these observations have implications that are comparableto the use of a non-associated flow rule (i.e. they exacerbatethe potential for instability), and may not apply to all classesof unsaturated geomaterials. For instance, if the water reten-tion curve depends significantly on the void ratio (termD71 6¼ 0), compressive deformation can promote a shift ofthe retention curve towards higher suctions (possibly havingbeneficial effects in terms of stability). At this point it isworth noting that, although consistent advances have beenproduced in describing the effect of volumetric strains onretention capabilities (Romero & Vaunat, 2000; Gallipoli etal., 2003), there is still little guidance for incorporating theeffect of shear strains (term D74). This fact complicatesevaluation of the interplay between retention properties andsoil stability. For these reasons, the effect of deformation onthe retention curve will not be accounted for in the follow-ing developments, thus using a simpler modelling strategy,one that is consistent with the limited geomechanical evi-dence available for collapsible unsaturated soils involved inflow slides.

APPLYING UNSATURATED SOIL MECHANICS TOSTABILITY OF SUB-AERIAL SHALLOW SLOPESStability indices for unsaturated shallow slopes

By following the strategy proposed by di Prisco et al.(1995), it is possible to derive stability indices in analyticalform and use them for the stability analysis of shallowdeposits. The extension of this procedure to unsaturatedslopes has recently been expounded in Buscarnera & diPrisco (2011b), and this paper is a numerical application oftheir analytical results. Hereafter, only some basic aspects ofthe theory are noted; the Appendix provides a description ofthe mathematical strategy used by the authors to derive thestability indices. Two triggering mechanisms are investi-gated: (a) a translational slide taking place under constantsuction (in this case permeability is assumed to be infinite);and (b) slope collapse initiated under water-content control(e.g. water-undrained shearing, water inundation, etc.). Thesefailure modes will be referred to as mode A and mode Brespectively.

Shear failure (mode A) can be considered the most usualform of material instability in slopes, and it is often thefailure mechanism included in conventional stability analysesfor unsaturated slopes (Ng & Shi, 1998; Gasmo et al.,2000). This mechanism is originated either by an increase inshear stresses or by a decrease in suction due to waterinfiltration. The former perturbation is conveniently repre-sented by a change in stresses at constant suction, and thelatter is often modelled through a decrease in suction atconstant total stresses. In both cases the control variablescoincide with those collected in the left-hand side of equa-tion (3). By following Buscarnera & di Prisco (2011b), astability index for this mechanism can be defined as

IAU ¼ D11D44 � D14D41 (6)

where the subscript U stands for unsaturated conditions. Theabove expression has been obtained by excluding singular-ities in the retention curve (i.e. D77 6¼ 0) and neglecting apossible role of strains in the retention behaviour (i.e.D71 ¼ D74 ¼ 0). Under these assumptions, condition IAU ¼ 0coincides with the strain localisation criterion obtained by diPrisco et al. (1995) for a saturated layer of an infinite slope,

IAS ¼ IAU (7)

where the subscript A refers to a shear failure mode, and Sstands for saturated conditions.

SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 3

The second triggering mechanism (mode B) occurs whenthe water content is controlled. This mode is relevant eitherwhen water drainage is prevented by natural layering orwhen water inlets from a surrounding formation can bemodelled as a fluid volume injected into the pores (Buscar-nera & di Prisco, 2011b). In either cases, changes in suctionare no longer imposed, but are obtained as an outcome ofthe deformation of the porous medium. From a mathematicalviewpoint this analysis is similar to passing from stress-controlled to strain-controlled conditions, and influences theonset of bifurcation. The stability index associated withfailure of the slope under constant water content has anexpression that is very similar to shear failure,

IBU ¼ D�11D44 � D14D�41 (8)

where the modified terms D�11 and D�41 reflect the role ofhydromechanical coupling in the considered failure mechan-isms (i.e. they also depend on degree of saturation, porosityand coupling terms D77, D17 and D47). The derivations ofD�11 and D�41, as well as their analytical expressions, aregiven in the Appendix. Buscarnera & di Prisco (2011b)showed that when Sr ¼ 0 (i.e. when the role of solid–fluidcoupling vanishes), IBU coincides with IAU, and the twoindices provide the same bifurcation mechanism. By con-trast, when Sr ¼ 1, the water-undrained deformation mode isnaturally associated with the initiation of static liquefaction.It is possible to expound this conceptual link by deriving thehydromechanical control matrix associated with the controlof total stresses and water content

_��

_�� _ew= 1þ eð Þ

8<:

9=;¼

D11 D14 n D77ð Þ�1D17� Sr

D41 D44 n D77ð Þ�1D47

Sr 0 n D77ð Þ�1

264

375

_��_ª��_s

8<:

9=;

(9)

in which e is the void ratio and ew is the water ratio, definedas ew ¼ eSr: Equation (9) can be used to reproduce theresponse of the slope when it is subjected to a hydromech-anical perturbation. The cases of water-undrained shearingand water inundation under dead load are included asparticular cases. It is possible to show that a singularity ofthe control matrix in equation (9) is governed by thestability index, equation (8) (i.e. its determinant vanisheswhen IBU ¼ 0).

When Sr ¼ 1, simple physical considerations allow equa-tion (9) to be rewritten to address the undrained loading of afully saturated soil. In fact, under saturated conditions,changes in the degree of saturation are no longer possibleand coupling effects disappear (i.e. D�1

77 ¼ 0, thus givingnD�1

77 D17 ¼ nD�177 D47 ¼ 0). In addition, since water is the

only pore fluid, the incremental response can be expressedas

_� �

_�� _ew= 1þ eð Þ

8<:

9=; ¼

D11 D14 �1

D41 D44 0

1 0 0

24

35 _��

_ª�� _uw

8<:

9=; (10)

in which the presence of the pore water pressure, uw, reflectsthe fact that the mechanical response is now governed bythe effective stresses. The condition involving changes inwater volume (i.e. _ew) has the role of enforcing the fluidmass balance (thus imposing an isochoric kinematics,_�� ¼ 0). It is straightforward to show that the control matrixin equation (10) vanishes when

D44 ¼ 0 (11)

Equation (11) coincides with the analytical condition forundrained failure under simple shear conditions (di Prisco et

al., 1995) and can be used to derive a stability index forstatic liquefaction,

IBS ¼ D44 (12)

where the subscript B indicates an undrained failure mech-anism under saturated conditions (subscript S).

It is therefore shown that the bifurcation mode associatedwith IBU ¼ 0 shares similarities with both shear strain locali-sation (indices IAS and IAU) and static liquefaction (indexIBS). The effect of terms D�11 and D�41 implies that, inparticular circumstances, mode B can occur before mode A.In other words, the hydromechanical constraint on thedrainage of water implies that, depending on the soil proper-ties, the features of this instability mode can be those ofeither a shear strain localisation or a liquefaction process(i.e. it can involve uncontrolled changes in pore waterpressures).

In the following, the particular features of these bifurca-tion modes will be elucidated by means of numericalsimulations. In order to simplify the comparison amongdifferent initial conditions and failure modes, the instabilityindices presented in this section will be reported in anormalised form, as

I�ij ¼I ij

I ij0

�� (13)

where Iij is a stability index and |Iij0| is a positive referencevalue that, unless otherwise stated, will be the absolute valueof the stability index for in situ conditions.

A constitutive model for unsaturated soilsThe theoretical framework presented in the previous sec-

tions can in principle be combined with any constitutive lawfor unsaturated soils. Since the focus of this paper is toinvestigate the stability of unsaturated slopes, the hydrome-chanical model developed by Buscarnera & Nova (2009) hasbeen selected. This constitutive law permits the use of anon-associated flow rule, and is formulated by means of amodelling strategy tailored to reproduce first-order featuresof unstable mechanisms in both saturated and unsaturatedsoils. This is achieved through constitutive functions that aredefined in terms of the so-called average skeleton stress

� 0ij ¼ � ij � Sruw�ij � 1� Srð Þua�ij (14)

Equation (14) reproduces the increase in stiffness andshearing resistance due to unsaturated conditions, as well asthe onset of shear failure resulting from saturation processes.Wetting paths, however, can originate inelastic effects muchbefore failure (a relevant example being the phenomenon ofwetting-induced compaction). A common strategy adopted tomodel these processes relies on the introduction of a depen-dence of the yield locus on suction and/or degree of satura-tion (Alonso et al., 1990; Jommi & di Prisco, 1994;Wheeler et al., 2003; Sheng, 2011). This peculiar feature ofunsaturated soils is reproduced here by including a hydrauliceffect in the hardening law, as

_p 0s ¼p 0s

Bp

_�pv þ �s _�p

s

� �� rswp 0s

_Sr (15)

where p 0s is the internal variable defining the size of the yieldsurface, �p

v and �ps are the volumetric and deviatoric plastic

strains respectively, and Bp, �s and rsw are hardening para-meters. The dependence of p 0s on the degree of saturationreproduces the expansion/contraction of the yield surfaceupon drying/wetting processes (Fig. 3), and implies couplingbetween the retention properties (here reproduced through an


uncoupled non-hysteretic Van Genuchten model; Van Gen-uchten, 1980) and the mechanical response of the material.Simulations of one-dimensional compression tests allow theimplications of these constitutive assumptions to be de-

scribed. Fig. 4(a) shows two stress paths, predicted forsaturated and unsaturated conditions respectively. Numericalsimulation of an increase in suction prior to one-dimensionalloading allows the effect of the parameter rsw to be shown.The expansion of the initial elastic domain postpones theonset of yielding, and reduces the amount of predicted plasticstrains upon loading. If wetting paths are eventually simu-lated, further plastic strains are predicted as a plastic compen-satory mechanism initiated by suction removal (Fig. 4(b)).Fig. 5 illustrates the role of some of the material constants inequation (15), showing that larger values of rsw and Bp areassociated with a larger potential for wetting collapse.

As was outlined in Buscarnera & di Prisco (2012), theprediction of hydromechanical instability requires an incre-mental formulation able to accommodate the notion ofmaterial stability. For this reason the incremental constitutiveequations of the model discussed in this section have beenarranged in accordance with equation (1). This choice allowsreinterpretation of the engineering problem pictured in Fig.

s1

s2

s3

p s0�

p s3� p s3� p s3�

p s3� p s2� p s1�

( , )s S3 r3

( , )s S2 r2

( , )s S1 r1

Suc

tion,

s

Degree of saturation,(a)

Sr

s1

s2

s3

Sr1 Sr2 Sr3

Wetting path

Suc

tion,

s

Internal variable,(b)

p s�

(saturatedconditions)

Mean skeleton stress,(c)

p �

Dev

iato

r st

ress

, q Wettingpath

Fig. 3. (a) Schematic description of retention curve and wettingpath; (b) hydraulic effects on evolution of internal variable p0s;(c) changes in size of elastic domain during wetting processes.Suction axes are to be considered in logarithmic scale

8070605040302010

100

0

10

20

30

40

50

60

70

80

0

Dev

iato

ric s

tres

s,

: kP

aq

Average skeleton stress, : kPa(a)

p �

Yield surface( 0·5)Sr �

Initial statebefore drying

( 1)Sr �

Oedometric stress path(unsaturated conditions)

Oedometric stress path(saturated conditions)

0

0·02

0·04

0·06

0·08

0·10

0·1210

Vol

umet

ric s

trai

n,ε v

Net vertical stress, : kPa(b)

σ netv

Unsaturatedconditions

Saturatedconditions

Wetting-induceddeformation

Fig. 4. (a) Initial elastic domains and predicted stress paths forone-dimensional compression under saturated and unsaturatedconditions (constitutive parameters given in Table 1);(b) predicted volumetric response for two simulations and effectof saturation path imposed at constant vertical net stress


1 (saturated conditions) for the case of unsaturated slopes(Fig. 6). Rainfall events imply, in fact, a variation of the insitu water pressure regime. Thus the distance from failureconditions can be defined in terms of changes in suction,˜s: that is, the external perturbation altering the state of theslope. The key issue is whether shear failure (˜ssf ) can beanticipated by other forms of collapse initiated by a wettingprocess (e.g. ˜swc in Fig. 6). In the following this logic willbe used to evaluate the distance from instability conditions.Although such an incremental definition of failure does notcoincide with the usual definition of safety factor, it hassome advantages that are specific to the present analysis(and more in general to the evaluation of instability condi-tions from coupled elasto-plastic soil models), as it allowsthe same strategy to be used for general types of instabilitymode.

Role of constitutive parameters in prediction ofhydromechanical bifurcation

Many components of the model can affect the capabilityof capturing unstable processes, such as the degree of non-

associativity, hydromechanical coupling (here introduced viathe parameter rsw), and the amount of inelastic wettingcompaction (governed primarily by the plastic compressibil-ity Bp). As previously pointed out, the role of couplingparameters in the prediction of these processes is unex-plored. In order to focus on a limited set of material proper-ties, and clarify their effect, it is useful to motivate theanalyses through an analytical inspection of the constitutiveequations. Consider for this purpose the stress–strain re-sponse for water-undrained isotropic loading (an example ofa stress path that involves changes in both stresses andsuction). If net stresses are controlled, and the loading pathis able to engage the plastic resources of the material, thenthe consistency condition requires

_p 0s ¼ _pnet þ _sSr þ s _Sr

¼ _pnet þ Ch_Sr

(16)

where Ch ¼ sþ (@ f R=@Sr)Sr is a coefficient that dependson suction, degree of saturation and retention curve (hereconsidered to be given by an expression of the types ¼ fR(Sr)).

By using equation (15), and assuming that plastic hard-ening involves only the volumetric strains (i.e. �s ¼ 0, as inclassical critical-state plasticity) it follows that

_pnet ¼ p 0s

Bp

_�pv � rswp 0s

_Sr � Ch_Sr (17)

0

0·02

0·04

0·06

0·08

0·10

0·12

0·14

0·16051015202530

Bp 0·038�

Bp � 0·060

Bp � 0·090

Vol

umet

ric s

trai

n,ε v

Suction, : kPa(a)

s

rsw 3·87�

Increase in plasticcompressibility ( )Bp

0

0·05

0·10

0·15051015202530

rsw 3·87�

rsw � 6·00

rsw � 8·00

Vol

umet

ric s

trai

n,ε v

Bp � 0·038

Suction, : kPa(b)

s

Increase in hydraulichardening parameter ( )rsw

Fig. 5. Parametric analysis: effect of (a) plastic compressibility(Bp) and (b) hydraulic hardening parameter (rsw) on thevolumetric strains predicted upon suction removal

Water infiltration ( )Δqw

α

ΔS

z

Soil

Bedrock

(a)

(b)

Δswc

Δssf

She

ar s

tres

s,τ

Normal skeleton stress, σ n�

In situ stressbefore rainfall

Shear failure locus

( * )s Sr 0

Fig. 6. (a) Schematic representation of rainfall infiltration (˜qw)acting over unsaturated infinite slope and causing perturbation ofsuction at material point level (˜s); (b) possible failure modes:difference in suction removal needed to activate shear failure(˜ssf ) or wetting collapse (˜swc)


During water-undrained loading paths, the changes in thedegree of saturation in equation (17) depend on the totalvolumetric strains (i.e. n _Sr ¼ Sr _�v). If plastic strains consti-tute the main contribution to the kinematic response (i.e._�v ’ _�p

v) it can be stated that

_pnet ’ p 0s

Bp

1� hð Þ _�pv (18)

where

h ¼ Bp

Sr

nrsw þ

Ch

p 0s

� �(19)

This simple analytical result shows that the presence ofhydromechanical coupling alters the mechanisms throughwhich the yield surface evolves. Changes in the degree ofsaturation, in fact, are no longer derived from imposedchanges in suction, but are rather a consequence of theentire mechanical response. As a result, the net stress rateassociated with plastic consistency (i.e. _pnet) is affected bythe pseudo-softening term h: Such hydraulic-induced soft-ening depends on retention properties (Ch), hydraulic hard-ening (rsw) and plastic compressibility (Bp). In particular, theparameter Bp exacerbates the potential for instability byamplifying the effect of the other hydromechanical terms.Since Bp is notionally associated with the expected amountof plastic strains originated by wetting (Fig. 5), theseconsiderations suggest that soils particularly prone to wettingcollapse tend to be characterised by a larger potential forhydromechanical instability.

The set of model parameters adopted for the numericalsimulations is given in Table 1. The parameters have beenselected with the goal of reproducing the response of a loose,unsaturated sand. The value and the range of variation of thebehavioural properties having a major effect on the stabilityindices (e.g. non-associativity, water retention parameters,hydromechanical coupling terms, etc.) have been assessed onthe basis of the available data for a volcanic silty sand(Bilotta et al., 2005; Buscarnera, 2010). Given the simplicityof the model, and the conceptual purpose of this paper,parametric analyses have also been performed. In particular,in order to disclose the outcome of an increasing potentialfor wetting collapse on the slope stability scenarios, theeffect of plastic compressibility has been explored in greaterdetail.

MODEL PREDICTION OF HYDROMECHANICALBIFURCATION MECHANISMSSimulation of material instabilities in unsaturated soils

In this section, simple shear simulations are used toillustrate the capabilities of the theory. A convenient startingpoint is the comparison between drained and undrainedshearing in saturated slopes. Fig. 7 shows two numericalsimulations characterised by the same initial conditions(slope angle Æ ¼ 108, � 9� ¼ 30 kPa). By using the inputparameters collected in Table 1, drained shearing produces aductile (strain-hardening) response, with the stability index

of shear strain localisation (equation (7)) converging asymp-totically to zero (shear failure). Once the perturbation modeis changed to undrained shearing, the stress path is charac-terised by a peak in shear stress (initiation of the instability).This is confirmed by the stability index for undrained simpleshear (equation (12)), the evolution of which is plotted inFig. 7(b).

Even though typical model predictions of drained shearingexhibit a stable mechanical response, alternative forms ofinstability can still be identified before critical state (i.e.before the stress state at which volumetric strains are nolonger possible). Fig. 8 details the evolution of the stabilityindices I�AS and I�BS during drained shearing. The figure

Table 1. List of parameters adopted for the numerical simulations

Elastic parameters Yield surface parameters Plastic potential parameters Hardening parameters SWCC parameters

Æ ¼ 0.00 af ¼ 0.63 ag ¼ 0.63 Bp ¼ 1/rs ¼ 0.038 aR ¼ 0.43k ¼ 0.02 mf ¼ 1.40 mg ¼ 1.40 �s ¼ 0.00 nR ¼ 1.3G0 ¼ 4000 kPa Mcf ¼ 0.9 Mcg ¼ 1.37 rsw ¼ 3.86 mR ¼ 0.22pr ¼ 100 kPa Mef ¼ 0.7 Meg ¼ 1.07

α �

3530252015105

: kPaNormal effective stress, σ ��(a)

0

5

10

15

20

25

She

ar s

tres

s,: k

Pa

τ �η

1·00·80·60·40·20

0

5

10

15

20

25

0

Drained shearing

Undrained shearing

She

ar s

tres

s,: k

Pa

τ �η

Shear failure locus

Shear failure

Peak in shear stress(initiation of instability)

In situ stress(after deposition)

10°

�0·2

(index for drained shearing)I *AS

(index for undrained shearing)I *BS

Normalised stability index(b)

Undrained instability( * 0)I BS �

Shear failure( * 0)I AS �

Fig. 7. Analysis of response of saturated slope via simple sheartests simulations: (a) stress paths for drained shearing andundrained shearing; (b) evolution of stability indices for drainedshearing and undrained shearing


shows that the condition I�BS ¼ 0 anticipates shear failure,defining a region in which a passage from drained toundrained conditions is critical. Since this passage is onlypotential, the onset of instabilities depends on the type ofshear perturbation. As a result, the region of the stress spacewhere I�BS < 0 corresponds to a domain of latent instability.Within such a region, model predictions suggest that flowinstabilities are possible, and their potential occurrence isnot overlooked if a convenient stability index is monitored.

The natural extension of these ideas to partially saturatedconditions relies on the comparison between constant-suctionand water-undrained shearing. This section details only thecharacteristics of model predictions obtained within the un-saturated regime; the next section will expound the unstabletransition from unsaturated to saturated conditions. In orderto show the role of material properties, water-undrainedshearing has been simulated by using two different values ofplastic compressibility (Fig. 9). While the constant-suctionscenario shares many similarities with the drained test pre-viously simulated, the water-undrained scenario may resembleeither drained or undrained shearing, depending on the initial

(index for drained shearing)I *AS

(index for undrained shearing)I *BS

Onset of latentinstability ( * 0)I BS �

Shear failure ( * 0)I AS �

Region oflatent instability

0

5

10

15

20

25

�0·2 0 0·2 0·4 0·6 0·8 1·0

She

ar s

tres

s,: k

Pa

τ �η

Normalised stability index

Fig. 8. Evolution of stability indices I�AS and I�BS during drainedshearing: concept of latent instability

α �

I * 0AU �I *BU �

(a)

srF �

(b)

Constant-suction shearing (CS )2Low compressibility ( 0·09)Bp �

Water-undrained shearing (WU )2Low compressibility ( 0·09)Bp �

She

ar s

tres

s,: k

Pa

τ �η

Initial suction( 0·5;

27 kPa)S

sr0

0

��

5040302010

α �

I * 0AU �

I * 0BU �

: kPaNormal skeleton stress, σ ��(c)

Stability index * (test CS )I AU 2

Stability index * (test WU )I BU 2

srF �

I *at constant suction)

AU � 0 (shear failure

I *at constant water content)

BU � 0 (instability

1·00·80·60·40 0·2

0

10

20

30

40

50Constant-suction shearing (CS )1Low compressibility ( 0·03)Bp �

Water-undrained shearing (WU )1Low compressibility ( 0·03)Bp �

She

ar s

tres

s,: k

Pa

τ �η

Shear failure locus


27 kPa)S

sr0

0

��

I * 0AU �

10°

0

10

20

30

40

50

0

10°

Stability index * (test CS )I AU 1

Stability index * (test WU )I BU 1

I *at constant suction)

AU � 0 (shear failure

I *at constant water content)

BU � 0 (instability

In situ conditions(after deposition)

0·55

�0·2

In situ conditions(after deposition)

Normalised stability index(d)

0·78

Fig. 9. Analysis of response of unsaturated slope via simple shear test simulations. Low/moderate compressibility: (a) stresspaths and (b) evolution of stability indices for constant-suction shearing and water-undrained shearing. Effect of highcompressibility: (c) stress paths for constant-suction shearing and water-undrained shearing; (d) evolution of stabilityindices. SrF indicates degree of saturation at moment of failure


state and material properties. If the material is characterisedby low or moderate compressibility, the evolution of theinstability index I�BU for water-undrained shearing is rathersimilar to that of constant-suction shearing, I�AU: In this casefailure is achieved on the same shear failure locus (Figs 9(a)and 9(b)), and the degree of saturation undergoes minorchanges. By contrast, high compressibility favours a responsethat is similar to the saturated/undrained scenario, withinstability initiating at lower shear stresses (Figs 9(c) and9(d)). In addition, in this case the degree of saturation under-goes more significant variations.

The results disclose a remarkable increase in complexitycompared to saturated conditions. The dependence of materi-al stability on the saturation index makes it impossible toestablish a direct correspondence between stress state andstability conditions. The latter depend on the incrementalloading path and the evolving state of the material, withlatent instability that tends to be predicted only for theloading paths that induce a non-negligible increase in thedegree of saturation. Under this viewpoint, the volume-change properties of the soil can be crucial. In fact, whenthe suction and the degree of saturation do not changeremarkably, suction-constant and water-undrained failuremodes tend to coincide (i.e. I�AU and I�BU vanish simulta-neously). By contrast, when soil compressibility is signifi-cant, Sr changes, and the water-undrained failure mode cananticipate suction-constant localisation (i.e. I�BU vanisheswhen I�AU is still positive). In this case the failure scenarioshares many similarities with the initiation of static liquefac-tion in saturated soils: it occurs when the soil has residualtendency to contract (i.e. when critical state has not yet beenreached), and implies a peak in shear stresses.

Stability indices can also support the analysis of unsatu-rated slopes when they are subjected to a more intuitiveform of perturbation: the saturation promoted by rain infil-tration. In this case, saturation tests with constant shearstress can be used to investigate stability conditions. In thiswork, the latter scenario is simulated by applying a removalof suction (˜s , 0) at constant net stress conditions. Fig. 10shows three simulations in which the saturation stage isimposed at different stress conditions, corresponding to threedifferent slope angles. The possibility of a localised shearfailure is checked first (Fig. 10(b)). At Æ ¼ 258 the value ofI�AU is always positive, given that the simulation does notexhibit any shear failure. Failure is reached at Æ ¼ 338 (i.e.very close to the angle of natural repose of the saturatedsoil), with the saturation index approaching zero at Sr ¼ 1.Finally, the simulation performed for a steeper slope angle,Æ ¼ 408, exhibits failure in the unsaturated regime, with I�AU

vanishing when Sr , 1.Even though the previous simulations are based on suc-

tion-controlled wetting paths (i.e. index I�AU governs theinitiation of failure), I�BU can still be monitored through alatent instability analysis. Fig. 11 refers to the simulation ofa suction-controlled saturation stage at Æ ¼ 308. Even ifshear failure is not attained during the process of suctionremoval, instability processes are still possible, depending onsoil properties. This is shown in Fig. 11(b), where I�BU ismonitored. The same simulation is in fact repeated by usingincreasing values of soil compressibility (i.e. by increasingthe tendency to produce volume compaction upon suctionremoval). The simulations show that highly collapsible soilscan suffer unexpected instability modes. The index I�BU

vanishes only for high values of soil compressibility, mark-ing the attainment of an unstable state. It is worth notingthat such predicted instabilities are only potential, beingcontingent to specific control conditions. In other words, themodel would predict an actual collapse of the slope onlywhen the system is perturbed in a certain manner (e.g., in

this case, by imposing water-undrained conditions or inject-ing water volume).

Unstable transition from unsaturated to saturated conditionsThis section discusses some particular types of model

prediction that are associated with the unstable transitionfrom unsaturated to fully saturated conditions. In otherwords, it illustrates a mechanical scenario that takes placeunder partially saturated conditions and provides a justifica-tion for catastrophic liquefaction failures. Fig. 12 illustratesa series of numerical simulations of water-undrained shear-ing starting from initially unsaturated conditions. Increasingvalues of compressibility are associated with larger changesin suction upon loading. This effect promotes a general shiftof the stress paths to the left, thus anticipating the predictionof instabilities.

The sequence of failure mechanisms can be clarified byinspecting the evolution of the stability indices. For rel-atively low values of plastic compressibility there is noprediction of local peaks in shear stress (simulations WU3

and WU4 in Fig. 12(a), and WU7 and WU8 in Fig. 12(b)),being shear failure achieved on the critical-state line(I�BU ¼ I�AU ¼ 0). Local peaks in shear stresses can instead

: kPaNormal skeleton stress, σ ��(a)

5040302010

1·11·00·90·80·70·60·5

0

10

20

30

40

50

0

Shear failure locus


27 kPa)S

sr0

0

��

Onset of shear failure

α � 25°

α � 33°

α 40°�

She

ar s

tres

s,: k

Pa

τ �η

0

0·2

0·4

0·6

0·8

1·0

0·4

Nor

mal

ised

sta

bilit

y In

dex,

*I AU

Degree of saturation,(b)

Sr

I * 0AU �

Initial degree ofsaturation in situ

α 25°�

α � 33°

α � 40°

Fig. 10. Simulation of the response of unsaturated slope by simpleshear tests: (a) stress paths for three saturation tests at constantshear stress; (b) evolution of stability index for failure underconstant suction (I�AU)


be predicted for larger values of compressibility (simulationsWU5 and WU6 in Fig. 12(a), and WU9 and WU10 in Fig.12(b)), and are associated with the fulfilment of I�BU ¼ 0prior to I�AU ¼ 0 (closed square symbols in Fig. 12). It isinteresting to note that the value of degree of saturation atfailure (SrF in Fig. 12) depends remarkably on the plasticcompressibility and the associated water-undrained stresspath, and should therefore be considered as a path-dependentcharacteristic.

If total stresses are assumed to be controlled, a peak inthe shear stress is associated with the prediction of anuncontrolled saturation of the pores. It is therefore particu-larly interesting to monitor the evolution of the index I�BS

during these simulations. In fact, since the post-peak volu-metric response keeps being contractive, continued shearingcauses a further increase in Sr and full saturation (opensquare symbols in Fig. 12). If saturation conditions areestablished, I�BS becomes the relevant index for undrainedshearing, and affects the predictions upon continued un-drained shearing. Thus very different types of post-peakresponse can be envisaged. In some cases the index I�BS isnegative at Sr ¼ 1 (simulations WU5, WU6 and WU9), thushaving a continued decrease in shear stresses. Other simula-tions are instead characterised by a recovery of shearingresistance upon continued shearing and two successive peaksin shear stress (simulation WU10). The latter circumstance

can be predicted if the saturation point is attained at rel-atively small values of mobilised friction angle (i.e. at shearstresses that do not yet correspond to spontaneous liquefac-tion). As illustrated in Fig. 13, this possibility implies arecovery in resistance (˜�), with a predicted strength cap-ability that can even overcome the shear stresses responsiblefor the activation of the first bifurcation mechanism. For thisreason, such simulated instability modes can be interpretedas predictions of metastable states – that is, situations atwhich the development of an unstable mode of deformationis interrupted by the transition from unsaturated to fullysaturated conditions.

Metastability can be also predicted during saturationpaths. Again, parallel monitoring of both the liquefaction-related stability index and the wetting-induced bifurcationcondition is important for identifying these mechanical con-ditions. Fig. 14 shows two simulations of saturation paths atconstant shear stress. While larger values of the slope angleimply that the stress threshold for incipient liquefaction iscrossed during wetting (I�BS ¼ 0 in Figs 14(a) and 14(c)),gentle inclinations are not associated with a state of incipi-

30°


5040302010

1·00·90·80·70·60·5

0

10

20

30

40

50

0

Shear failure locus


27 kPa)S

sr0

0

��

I * 0: Latent instabilityBU�

She

ar s

tres

s,: k

Pa

τ �η

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

sta

bilit

y In

dex,

*I BU

Degree of saturation,(b)

Sr

Initial degree ofsaturation

α �

Latent instability: I * 0BU �

Bp 0·03�

Bp � 0·05Bp � 0·07Bp � 0·09

Fig. 11. (a) Possibility of latent instabilities during saturation;(b) evolution of stability index for wetting collapse (I�BU)


161412108642

α �

WU5

Bp � 0·08

SrF 0·93�

WU4

Bp � 0·06

SrF 0·89�

WU3

Bp � 0·04

SrF 0·8�

She

ar s

tres

s,: k

Pa

τ �η

0

2

4

6

8

10

12

14

16

0

She

ar s

tres

s,: k

Pa

τ �η


7·5 kPa)S

sr0

0

��

ConditionSymbols

I * 0AU �

I * 0BU �

I * 0BS �

Sr � 1

I * 0at saturation

BS �

WU6

Bp � 0·10

SrF 0·87�

15°

0

5

10

15

20

25

0


7·5 kPa)S

sr0

0

��

15°α �

WU9

Bp � 0·08

SrF 0·96�

WU8

Bp � 0·06

SrF 0·93�

WU7

Bp � 0·04

SrF 0·8�

WU10

Bp � 0·10

SrF 0·87�

I * 0BS �

I * 0BS �

252015105: kPaNormal skeleton stress, σ ��

(b)

Fig. 12. Simulation of water-undrained simple shear mechanisms:effect of soil compressibility on unstable transition fromunsaturated to saturated conditions


ent liquefaction (I�BS . 0 in Figs 14(b) and 14(d)). Thisdifference is fundamental in grasping the effects of wettingcollapses initiated by suction removal. While an unstabletransition from unsaturated to saturated conditions is thetrigger of a subsequent liquefaction in the first case (I�BS , 0when Sr ¼ 1 in Fig. 14(e)), this is not the case in gentleslopes, given that possible wetting collapses are transientmetastable conditions, not necessarily associated with a sub-sequent liquefaction of the layer (i.e. shearing resistance canbe recovered after the first bifurcation mechanism; Figs14(b) and 14(f)).

STABILITY CHARTS OF HYDRAULICPERTURBATIONS: EFFECT OF SUCTION REMOVAL

The stability of unsaturated deposits during rainfall eventscan be studied by simulating the response of the slope towetting paths. Although the complete quantification of suc-tion perturbations over time (as well as the associated rain-fall thresholds for slide triggering) would require data fromtransient rain infiltration analyses, it is possible to simplifythe description of the hydrologic effects by representing theirperturbations as suction removal processes (i.e. ˜s , 0). Inthis way the changes in suction are closely related to thedisturbance effectively altering the current state of the slopeduring a rainfall event, and are likely to be associated withthe onset of material failure and the consequent activation ofa slide.

The material point simulations illustrated in the previoussection can be used to condense the effect of materialparameters and slope characteristics (e.g. deposit thickness,slope inclination). In other words, the perturbations able toinduce an instability can be identified through the stabilityindices obtained from material point analyses. Such simula-tions are therefore used to define instability scenarios forgiven sets of initial saturation conditions, slope inclinationsand types of disturbance. The outcome of these analyses iseventually collected in graphical charts of triggering pertur-bations, hereafter referred to as ‘stability charts’. It is worthnoting that, although the focus of this paper is on themechanical implications of suction removal processes, such

a strategy can in principle be combined with more sophisti-cated retention models and with advanced hydrologic de-scriptions of the infiltration process, therefore studying theeffect of prior infiltration/evaporation events.

The stability charts discussed in this section will be basedon suction-controlled wetting tests, and will be presented interms of changes in suction. In order to compare differentinitial conditions and have a more convenient graphicalrepresentation, triggering perturbations are normalised forthe initial suction, s0: In this way, full saturation is achievedwhen the normalised perturbation ˜s� ¼ ˜sj j=s0 ¼ 1:

The possibility of shear failure induced by saturation pathsis investigated first. Fig. 15 shows the stability chartsobtained by studying the evolution of I�AU for different initialsaturation conditions. All charts converge to the same point,coinciding with the angle of natural repose (ÆNR) of thesaturated material. For that inclination, in fact, failure isobtained only when suction is completely removed. Thestability charts are not defined in the range of slope angleslower than the angle of natural repose, for which thecontribution of suction is not necessary to ensure stability.By contrast, the value of suction is critical to assess stabilityconditions when Æ . ÆNR, as higher suctions enable thematerial to sustain steeper inclinations. Fig. 15(b) illustratesthe practical use of these stability charts. Once the in situstate is defined, the associated chart provides the magnitudeof triggering perturbation for any slope inclination. It is thenpossible to evaluate this critical change in suction for adeposit of a given inclination.

Shear failure is the most intuitive type of instabilityexpected during saturation. As previously shown, however,other forms of instability are possible upon suction removal.This possibility is explored in Fig. 16, where stability chartsof the index I�BU are reported. Any point of the chart isassociated with condition I�BU ¼ 0: Thus the charts havebeen obtained by controlling the evolution of the stabilityindex associated with water-content control during the pro-cess of suction removal. As in the previous examples, theeffect of soil compressibility is investigated. The effect ofthis parameter on the stability charts is remarkable. Whereasfor relatively stiff soils there is practically no difference withthe stability chart of shear failure, higher values of compres-sibility alter significantly the predicted stability scenario.The charts derived from I�BU indeed tend to shift below thoseassociated with I�AU: In other words, the numerical resultsreflect the fact that the state of the slope has entered adomain of latent instability, and suggest that a multiplicityof failure modes can be predicted for highly collapsiblematerials (i.e. volumetric instabilities can arise before theattainment of shear failure). These considerations can belinked to the notion of metastability discussed in the pre-vious section. Depending on the specific material param-eters, the prediction of mathematical bifurcation can indeedassume different connotations, which in turn reflect distinctforms of wetting instability. As was indicated in the previoussection, information on metastable states must be derivedfrom the combined analysis of indices I�BU and I�BS: Forexample, for high values of compressibility the stabilitychart associated with I�BU ¼ 0 shifts below the chart ofI�BS ¼ 0 (Fig. 17(b)). This circumstance reflects the possibil-ity of a metastable condition (as illustrated in Fig. 14(b)),and does not suggest a spontaneous sequence of unstablemechanisms. On the contrary, when the slope angles proneto latent instability lie within the range of potential liquefac-tion (i.e. I�BS < 0), the model predicts the possibility of twosuccessive instability mechanisms leading to liquefaction(Fig. 17(a)). In other words, if I�BU vanishes when thestability boundary for flow failure has already been crossed,wetting instability should be regarded as a precursor for

Δτ

Firstbifurcation:

* 0I BU �

30252015105

: kPaNormal skeleton stress, σ ��

Ss

B

r0

0

p

0·77·5 kPa

0·10

�

�

�

0

2

4

6

8

10

0

She

ar s

tres

s,: k

Pa

τ �η

Saturationof the pores:

* 0I BS �

Secondbifurcation:

* 0I BS �

Fig. 13. Numerical prediction of metastable conditions: first peak(bifurcation under unsaturated conditions, I�BU 0) causes fullsaturation, and is followed by second bifurcation point (secondpeak at I�BS 0) at larger stresses (˜� indicates the predictedrecovery in shearing resistance)


liquefaction, given that the initiation of unstable saturation isfollowed by a continuous reduction in shearing resistance. Inthe light of these analyses, the achievement of fully saturatedconditions can be the consequence of an unstable processrather than a prior cause of collapse, and the liquefactionevent can be understood as the ultimate stage of a chainprocess activated by a prior suction removal.

The theoretical interpretation of the numerical simula-tions allows two opposite scenarios to be distinguished. Thefirst one refers to soils whose volumetric response is ratherinsensitive to wetting paths. In this case, there is nopractical difference between I�AU and I�BU (both providingthe same prediction of failure; Fig. 18(a)), and shear strainlocalisation is the only mechanism that can be initiated by

α �

I * 0AU �

I * 0BU �

I * 0BS �

Sr 1�

1412108642: kPaNormal skeleton stress, σ ��

(a)

Initial suction( 0·7; 7·5 kPa)S sr0 0� �

Change incontrol

Attainment offull saturation

α �

3530252015105

: kPaNormal skeleton stress, σ ��(b)

α � α �33° 15°

I *AU

I *BU

I *BS

1·000·950·900·850·800·75 1·000·950·900·850·800·750·70

Degree of saturation,(d)

Sr

I *BS

I *AU

I *BU

α � α �33° 15°

�1·0

�0·5

0

0·5

1·0

1·5

Nor

mal

ised

sta

bilit

y in

dice

s, *I

1·000·950·900·850·800·75 1·000·950·900·850·800·750·70

Degree of saturation,(f)

Sr

0·70

Degree of saturation,(e)

Sr

�0·4

�0·2

0

0·2

0·4

0·6

0·8

1·0

1·2

I *BUI *BS

I *BS

I *BU

0

2

4

6

8

10

12

14

0

She

ar s

tres

s,: k

Pa

τ �η

ConditionSymbols

Change incontrol

Attainment offull saturation

Shearfailure


7·5 kPa)S

sr0

0

��

P

33°

0

5

10

15

20

0

P

15°

�1·0

�0·5

0

0·5

1·0

1·5

0·70

Degree of saturation,(c)

Sr

I *BS � 0 at P

P (change in control)

Nor

mal

ised

sta

bilit

y in

dice

s, *I

I *BU � 0 at P

�0·4

�0·2

0

0·2

0·4

0·6

0·8

1·0

1·2


I *BS � 0 at P

I *BU � 0 at P

Water-undraineddeformation stage


I *BS �

�

0at 1Sr


Water-undraineddeformation stage

I *BS � �0 at 1Sr

Fig. 14. Numerical simulation of (a) unstable transition from unsaturated to saturated conditions and(b) metastable response (Bp 0.10); (c), (d) evolution of three stability indices during saturation paths;(e), (f) evolution of I�BU and I�BS after onset of water-undrained shearing (point P). Dotted lines in (e) and (f)indicate evolution of I�BU and I�BS prior to change of control


suction removal. This scenario is possible for slope angleslarger than the angle of natural repose of the saturatedmaterial (interval 2 in Fig. 18(a)). The second scenariorefers to soils exhibiting significant volume collapse uponwetting. In this case, hydromechanical coupling can becritical, and yields the existence of additional instabilitymodes. The condition I�BU ¼ 0 originates a distinct stabilitychart, located below the stability domain associated withI�AU ¼ 0 (Fig. 18(b)). The extent of the range of slopeangles that are either unaffected by instability mechanisms(interval 1) or which suffer only shear failure (interval 3)becomes smaller. In contrast, an additional range of criticalslope angles is found (interval 2), for which wettingcollapse can dominate the failure response of the slope.The main outcome of this scenario is an extension of therange of unstable slope angles and, most notably, a changein the features of the expected instability mechanism. Thepredicted wetting-instability modes can be further differen-tiated on the basis of the value of the index I�BS forsaturated-flow failure (Fig. 19). In all cases, the onset of awetting instability under total stress control (I�BU ¼ 0) hasthe effect of inducing full saturation (i.e. ˜s� ¼ 1, asillustrated by the vertical arrows in Fig. 19). After thisprocess, different scenarios can be devised. In particular,when the stability charts obtained from the index I�BS

indicate that the boundary for a possible isochoric flow arecrossed before the fulfilment of condition I�BU ¼ 0, thewetting instability takes place within a domain that isalready prone to static liquefaction (i.e. I�BS < 0). In thiscase, wetting instability can be regarded as the hydrome-chanical trigger of a flow instability. By contrast, when theunstable saturation of the pores coincides with I�BS > 0, itreflects the existence of a metastability domain in whichcondition I�BU ¼ 0 is no longer a precursor of catastrophicliquefaction (interval 2a in Fig. 19(b)).

CONCLUSIONSThis paper has detailed the study of flow slides triggered

by rainfall. The aim has been to provide a modelling frame-work for explaining failure in unsaturated slopes and per-forming triggering analyses. For this purpose, the scheme ofan unsaturated infinite slope has been combined with thebasic principles of unsaturated soil elasto-plasticity and themathematical concept of controllability. In this way, theresponse of the deposit has been reproduced by means ofsimple shear simulations, and analytical indices have beenused to study the stability of the slope.

Sr0 0·8�

s0 4·0�

Sr0 0·9�

s0 1·75�

Sr0 0·7�

s0 7·50�

Sr0 0·6�

s0 14·0�

70605040302010

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0

Slope angle, : degrees(b)

α403530252015105

Sr0 0·8�

In situ state:35°

35 kPa0·8

4·0 kPa

α ��

��

γzS

sr0

0

0

0·2

0·4

0·6

0·8

1·0

0

Increase in Sr0No localisedshearing failureupon saturation

αNR

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0

γz 10 kPa�

Slope angle, : degrees(a)

α

0

αNR

No localisedshearing failureupon saturation

I * 0: shear failureAU �upon suction removal

Fig. 15. Stability charts of hydraulic perturbations for shearfailure (charts obtained by checking occurrence of I�AU 0:(a) role of initial saturation conditions; (b) example of use ofchart for given in situ conditions

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0


α403530252015105

In situ state:35°0·8

4·0 kPa

α ��

�S

sr0

0

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0

0

Ss

r0

0

0·84·0 kPa�

�35 kPaγz �


α4035302520151050

Ss

r0

0

0·84·0 kPa�

�35 kPaγz �

Bp 0·03�

Bp � 0·05Bp � 0·07Bp � 0·08

Bp 0·03�

Bp � 0·05Bp � 0·07Bp � 0·08

Increase in soilcompressibility

I I* *AU BU� � 0: shear failureupon suction removal

I * 0: possible latent instability(volume collapse can take place

upon suction removal)

BU �

Fig. 16. Stability charts of hydraulic perturbations for latentinstability during suction-controlled saturation (charts obtainedby checking occurrence of I�BU 0). Role of soil compressibility inthe possibility of entering a region of latent instability


The analyses show that wetting paths can trigger a multi-plicity of unstable phenomena, and that some of theseinstabilities can anticipate shear failure. Three types ofmechanism have been studied: localised shear failure, staticliquefaction and wetting collapse. In particular, it has beenshown that the unstable mode associated with wetting-induced collapse shares several features with static liquefac-tion. The major difference, however, is that wetting-collapsephenomena are predicted to occur when the material is notyet saturated, and can therefore be activated by the processof suction removal. According to this interpretation, satu-rated conditions may not be necessary to trigger a flow slide,being liquefaction potentially originated from a chain pro-cess consisting of volume collapse, uncontrolled saturationand, eventually, catastrophic undrained failure. These ana-lyses point out that the combined use of several stabilityindices is critical for distinguishing different failure scenar-

ios. They also emphasise the importance of using a unifiedstrategy of analyses, in which shear failure, saturation-induced liquefaction and metastability are all naturally in-cluded as particular cases.

In order to highlight the engineering significance ofthese notions, stability charts representing the triggeringperturbations as a function of the slope inclination havebeen numerically evaluated. The analyses show that thepossibility of undergoing volumetric instabilities also de-pends on the parameters that introduce hydromechanicalcoupling. This conclusion suggests that rainfall-inducedflow slides are exceptional phenomena that can take placeonly in very particular deposits, susceptible to both under-going liquefaction and experiencing volume compactionupon saturation. Most notably, these results suggest thatthe risk of rainfall-induced flow slides may depend onmaterial properties that are not directly associated with theshearing resistance. In order to clarify these concepts,parametric analyses have been performed. Such analysesallowed assessment of the relation between the parametersgoverning wetting-induced compaction and the range ofslope inclinations affected by the instability mechanisms. Ithas been found that the instability of slopes made ofmaterials that are insensitive to wetting paths is dominatedby shear failure, liquefaction being possible only underfully saturated conditions. In contrast, soils that are highlycollapsible upon wetting are associated with a much broad-er spectrum of unstable responses, which also includes theinitiation of liquefaction upon either suction removal orsaturation-induced metastability.

40353025

40353025200

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0


α

0

0·2

0·4

0·6

0·8

1·0

Nor

mal

ised

suc

tion

pert

urba

tion,

*|

|/Δ

Δs

ss

�0

Ss

r0

0

0·84·0 kPa�

�35 kPa

0·07γz

B�

�p


α

Ss

r0

0

0·84·0 kPa�

�35 kPa

0·08γz

B�

�p

Δs Ifor * 0AU �

Δs Ifor * 0BU �

Δs Ifor * 0BS �

20

Range of slope anglesfor wetting instability:I I* 0 and * 0BU BS� �

Range of slope anglessusceptible to metastability:

* 0 and * 0I IBU BS� �

Δs Ifor * 0AU �Δs Ifor * 0BU �

Δs Ifor * 0BS �

Fig. 17. Stability charts for wetting instability (I�BU 0) andliquefaction (I�BS 0): (a) stability threshold for incipient lique-faction already crossed at I�BU 0 (unstable transition fromwetting instability to flow failure); (b) at low angles I�BU 0anticipates conditions of incipient liquefaction (I�BS > 0 andpossible metastability is predicted at Sr 1)

1

0

||/

Δs

s 0

1 2

(a)α

Localisedshear failure

* 0I *AU � I BU �

Shearfailure* 0I AU �

1

0

||/

Δs

s 0

1 2

(b)α

3

Latent instabilityI * 0BU �

Fig. 18. Schematic representation of stability charts. (a) Water-insensitive soils: shear strain localisation is only failure mode (I�AU

and I�BU provide the same results). (b) Relevant tendency tocollapse on wetting (hydromechanical coupling). Latent instabilityis distinguished from shear strain localisation (I�AU and I�BU do notprovide the same results)


Although many other factors that can affect hydromech-anical stability (e.g. coupled retention properties, hydraulichistory) have not been specifically addressed in this paper,the theoretical methodology can be enhanced for quantify-ing their effect. At variance with the usual stability analysesfor unsaturated slopes, the proposed approach is based onstability indices that reflect the role of suction removalprocesses and enable multiple failure mechanisms to besimultaneously accounted for. As a result, in a futureperspective this theory can be combined with detailedexperimental characterisation of unsaturated deposits andhydromechanical models calibrated for site-specific features.In the authors’ opinion, the achievement of such a coordi-nated effort between geomechanical modelling and geotech-nical site characterisation can constitute a powerful tool forlocating areas prone to originate slope failures and estimat-ing the risk of activation of flow slides.

APPENDIXThis section briefly illustrates the analytical derivation of the

stability indices used in this paper. More details are available inBuscarnera & di Prisco (2011b). Equation (20) reports the completeincremental hydromechanical response of the material points withina slope.

_� ��_� ��_� �_��

_��

_��

_s�

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

¼

D11 D12 D13 D14 D15 D16 D17

D21 D22 D23 D24 D25 D26 D27

D31 D32 D33 D34 D35 D36 D37

D41 D42 D43 D44 D45 D46 D47

D51 D52 D53 D54 D55 D56 D57

D61 D62 D63 D64 D65 D66 D67

D71 D72 D73 D74 D75 D76 D77

2666666666666664

3777777777777775

_��

_��

_�

_ª��

_ª�

_ª�

� _Sr

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;(20)

The kinematic constraints deriving from the assumption of aninfinite slope can be used to simplify the above relations. Plane-strain conditions imply that _� ¼ _ª� ¼ _ª� ¼ 0, while the sym-metry along the � axis implies _�� ¼ 0: These kinematic constraintsare representative of a simple shear strain mode, and lead toequation (3).

The stability index for mode A can be obtained by considering anincremental loading path characterised by controlled changes instresses and suction (e.g. constant-suction shearing, constant-stresssuction removal). In this case the control variables coincide withthose collected in the left-hand side of equation (3). The theory oftest controllability therefore identifies the inception of a bifurcationmode when the constitutive matrix in equation (3) is singular: that is,when

D77 D11D44 � D14D41ð Þ ¼ 0 (21)

in which conditions D71 ¼ D74 ¼ 0 have been used.Mode B can be derived by modifying equation (3) to reproduce

water-undrained loading. If the drainage of the water phase isprevented, the evolution of both degree of saturation and suctiondepends on the overall mechanical response of the material. Underconstant water content, the constraint relating volumetric strains andsaturation index is given by

n _Sr ¼ Sr _�� (22)

By expressing _�� as a function of the increment in total normalstress and suction (i.e. _�� ¼ _�� þ Sr _s), the constitutive equationscan be reformulated as

_� �

_��

¼ D�11 D14

D�41 D44

� �_��_ª��

(23)

where

D�11 ¼ D11 �Sr

nD17 �

Sr

nD77

� �(24)

D�41 ¼ D41 �Sr

nD47 (25)

in which the hydraulic variables are eliminated by using theconstraint in equation (22). Equations (23) are complemented byn_s ¼ �(Sr=n)D77 _��, which is needed for tracking the changes insuction during shearing. It is interesting to note that the stabilityindex associated with failure of the slope under constant watercontent coincides with the determinant of the control matrix inequation (9).

NOTATIONaf shape parameter of the yield surfaceag shape paremeter of the plastic potentialaR shape parameter of the retention curveBp plastic compressibility parameterCh hydraulic parameter related to retention propertiesD hydromechanical constitutive matrix

Di j principal minors of the constitutive stiffness matrix_E hydromechanical strain vectore void ratio

ew water ratiofR soil water retention curve

G0 elastic shear modulusIAS stability index for shear failure (saturated soil)IAU stability index for shear failure (unsaturated soil)IBS stability index for liquefaction (saturated soil)

Saturation instability ( 1)activates liquefaction

Sr �

1

0

||/

Δs

s 0

Staticliquefaction

* 0I BS �

2

α(a)

Latent instability* 0I BU �

Saturation instability ( 1)Sr �

1

0

||/

Δs

s 0

* 0I BS �

2a

α(b)

Latent instability* 0I BU �

2b

Metastability Liquefaction

Zone ofpossible

metastableevolution

Fig. 19. Schematic representation of instability charts. (a) Stabi-lity chart for I�BS 0 is entirely located below condition forI�BU 0; static liquefaction can be the ultimate consequence of awetting instability. (b) Stability chart for I�BU 0 can be above thecondition for I�BS 0 (interval 2a); evolution of wetting instabilitycan be characterised by condition of metastability (shaded areaindicates zone of possible metastable evolution)


IBU stability index for liquefaction (unsaturated soil)I�ij normalised stability indices

k elastic compressibilityMcf shape parameter of the yield functionMcg shape parameter of the plastic potentialMef shape parameter of the yield functionMeg shape parameter of the plastic potentialmf shape parameter of the yield functionmg shape parameter of the plastic potentialmR shape parameter of the retention curve

n porositynR shape parameter of the retention curvepr reference pressure for nonlinear elasticity

_pnet net stress incrementp 0s internal variable for isotropic hardening

˜qw rainfall infiltrationrsw hydraulic hardening parameterSr degree of saturation

SrF degree of saturation at failures suction

_s� rate of smeared suction˜s perturbation of suction

˜ssf triggering suction perturbation for shear failure˜swc triggering suction perturbation for wetting collapse˜s� normalised triggering suction perturbation

ua pore air pressureuw pore water pressure

d2W second-order work per unit volumeX constitutive control matrixz vertical depthÆ slope angleªz vertical overburden�ij Kronecker’s delta�ij total strain tensor�p

v volumetric plastic strain�p

s deviatoric plastic strains�e

ij elastic strain tensor�p

ij plastic strain tensor� axis of the reference system (tangential to the slope)� axis of the reference system (normal to the slope)�s dilatancy hardening parameter_� hydromechanical stress vector

�ij total stress tensor� 9ij effective stress tensor� 0ij skeleton stress tensor

_� �ij incremental skeleton stress

� 9n in situ normal effective stress˜� shear perturbation

˜�liq triggering shear perturbation for liquefaction˜�sf triggering shear perturbation for shear failure

� vector of control variablesh hydraulic-softening parameterł vector of response variables

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