instability of unsaturated soils: a review of theoretical ... · pdf fileinstability of...

Journal of Geo-Engineering Sciences 2 (2014) 39–65DOI 10.3233/JGS-130012IOS Press

39

Instability of unsaturated soils: A reviewof theoretical methods

Giuseppe Buscarnera∗ and Constance MihalacheDepartment of Civil and Environmental Engineering, Northwestern University, Evanston, IL, USA

Abstract. The paper reviews a series geomechanical approaches for interpreting instabilities in unsaturated geomaterials, a classof solids involved in numerous geo-engineering problems, such as hazard forecasting, infrastructure management, undergrounddisposal of by-products and energy technologies. The review details the connection between second-order work input, loss ofcontrollability and material failure. Hydro-mechanical problems are addressed, focusing on a specific class of environmentalperturbations that can cause sharp changes in both mechanical and hydrologic variables. A procedure to define the second-orderwork input to an unsaturated soil volume is discussed first. It is pointed out that this energy measure motivates the incrementalform of the constitutive laws for stability analyses. It is then discussed how to link the theory of hydro-mechanical controllabilitywith constitutive approaches for unsaturated soils, which are typically based on the framework of strain-hardening plasticitywith extended hardening. In doing so, the crucial role of the properties that govern the interactions between soil skeletonand fluid-retention processes is emphasized. The implications of these findings are commented with reference to a specificapplication: the forecasting of landslide triggering in natural slopes. It is shown that the use of suitable stability indices allowsone to differentiate between frictional slips and volumetric collapses turning into flows. These results suggest that geomechanicaltheories calibrated for site-specific properties can support the quantitative assessment of landslide susceptibility, as well as anumber of other engineering applications involving the instability of unsaturated porous media.

Keywords: Unsaturated soils, hydro-mechanical coupling, second-order work, material stability, bifurcation

1. Introduction

Geomaterials are porous, discrete and inherently heterogeneous media that are continuously exposed to interac-tions with the surrounding environment [50]. These natural solids host multiple fluids, with which they establishphysical and chemical interactions. Natural and human-induced fluctuations in environmental conditions have theability to change the physical properties of these media, thus playing a primary role in a number of applications,such as the forecasting of natural hazards, the management of aging infrastructures, the optimization of energy-production technologies and the underground storage of hazardous substances (e.g., nuclear waste or liquefiedcarbon dioxide) [2, 22, 58, 74, 111].

This review focuses on a specific class of environmental processes: the hydro-mechanical interactions at failurebetween the solid skeleton and the fluids hosted within its pores. More specifically, the paper aims to envisage arange of methods to detect potential instabilities in unsaturated soils, as well as in the earthen systems made ofthem. The main goal is to show that the general problem of material stability in such class of natural media canbe addressed though a set of concepts similar to those that have characterized stability analyses in other types of

∗Corresponding author: Giuseppe Buscarnera, Department of Civil and Environmental Engineering, Northwestern University, Sheridan Road,Evanston, IL 2145, USA. E-mail: [email protected].

ISSN 2213-2880/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

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40 G. Buscarnera and C. Mihalache / Instability of unsaturated soils: A review of theoretical methods

solids, such as metals [42, 43, 59], cohesive geomaterials [9, 79] or fluid-saturated soils [11, 69, 87]. Indeed, thecharacterization of strength properties and failure modes still represents a challenge for geomechanics, especiallyfor the case of unsaturated soils, whose strength, deformability and stability are invariably linked with the state oftheir pore fluids.

The interest in understanding failure events in geomaterials has generated a wide range of methods for interpretingexperiments on soils and rocks (extensive reviews of these approaches are available, for instance in Vardoulakis andSulem [109], Darve and Vardoulakis [37] and Besuelle and Rudnicki [8]). Most of the available methodologies forgeomaterial stability derive from the pioneering studies by Drucker [42] and Hill [59], that are usually linked to theprinciple of effective stresses [106] and specialized to saturated geomaterials [35, 68, 70, 86]. Along these lines,many recent works have highlighted the usefulness of second order-work criteria to capture mechanical instabilitiesof relevance for natural slopes [e.g., 25, 28, 30, 80]. Indeed, the combination of mechanical theories with predictiveconstitutive models guarantees remarkable capabilities in capturing unstable events in both saturated and unsaturateddeposits at laboratory and field scales [19, 21, 36, 40].

When unsaturated soils are considered, however, it is readily apparent that the enhancements of experimental andconstitutive approaches [38, 51, 101] have not been paralleled by comparable advancements of the stability criteria.Indeed, most approaches used to back-analyze failure in unsaturated soils adopt methods conceived for saturatedsoils, the most frequent adaptation being related to the use of enhanced effective stress measures and/or suction-dependent strength [47, 67, 77, 89]. Although the definition of a stress measure is a crucial step for unsaturatedsoil modeling, it is not by itself sufficient to address the general problem of constitutive stability, as the latter isinvariably governed by energy interactions among different phases [31, 52, 98]. Such interactions are indeed notcaptured by a single effective stress measure [17, 52, 60] and require extended stability indicators. In addition, theassessment of stability conditions must consider the inelastic processes triggered by changes in degree of saturation,which usually anticipate frictional failure. Such inelastic mechanisms are critical to elucidate the nature of wetting-induced collapses [6, 61, 65, 104], a class of deformation processes rarely framed within fundamental physicaltheories and/or explained though convincing mechanical arguments [76]. Saturation induced-collapses assume aquite ambiguous connotation also in other contexts, such as the initiation of flow failures due to water infiltration[72, 90, 91, 103]. Extensive studies have shown that the presence of gas within the pore fluids, and hence of astate of partial saturation, alters the threshold for liquefaction instability, both under monotonic and cyclic loadingconditions [39, 53, 73, 97]. In particular, small amounts of gas can significantly reduce the potential for solid-to-fluidtransitions. These observations have inspired a variety of methodologies for mitigating liquefaction hazards [57, 92,93], but have been rarely complemented by the theoretical analysis of the effect of partially saturated conditions onthe instability thresholds, nor by comprehensive studies of the role of environmental fluctuations (e.g., tidal waves,sudden gas release, fluid injection/extraction processes, infiltration events, etc.), which in this context can be theprimary cause of unexpected failures [16, 17, 72].

By analogy with collapse phenomena in fluid-saturated soils, wetting collapses and saturation-induced solid-to-fluid transitions are often treated as diffuse modes of failure. On this matter, however, several fundamental questionscan be asked. What is indeed the role of the degree of saturation in the alteration of the geometric patterns of afailure mode? Are changes in saturation conditions able to promote a transition from diffuse to localized modes offailure (or vice versa)? Several studies on unsaturated soils show that the localization of strains into planar zones ofintense deformation are indeed quite common in unsaturated soils [33, 34, 62]. This has been observed either thoughbiaxial experiments [32, 62] or by simulating localization processes through advanced numerical techniques [13,24, 45, 96]. Although these observed and/or simulated localization processes exhibit phenomenological patternsvery similar to those seen in saturated media [55, 82], experiments and simulations are rarely inspected in light ofbifurcation criteria specialized to unsaturated conditions.

Notable exceptions can be tracked back to the works by Borja [10], who identified differences between thelocalization conditions for fully drained and locally undrained shear banding, and by Vaunat et al. [110], whosuggested the use of the bifurcation theory to detect non-homogeneities in the water content distribution. Theseideas contain the seeds of a potential adaptation of bifurcation theories to unsaturated soils. The current reviewstarts from these inspiring notions, and tries to make a fresh start in the analysis of instabilities and bifurcationmodes in unsaturated soils. The aim of the paper is to outline a unified framework of interpretation that extends the

G. Buscarnera and C. Mihalache / Instability of unsaturated soils: A review of theoretical methods 41

concepts of second-order work, controllability and bifurcation to this class of natural media. The methods that will bedescribed in the following section will rely on the mathematical concepts of incremental bifurcation, uniqueness andexistence of the mechanical response. Indeed, since the observation of failure in experiments is always subjectedto interpretation, the assessment of failure conditions must be based on precise theoretical statements, that canbe eventually used to specify mechanical criteria and back-analyse failures [109]. With this goal in mind, thepaper summarizes some of the recent works proposed by the first author together with many other collaborators[16, 17, 22, 23, 81], and discusses how these fundamental mechanical notions relate with the state of saturation,the static/kinematic conditions, the pressures of the pore fluids and the perturbation of the hydrologic variables. Thegeneral mechanical problem is presented first, with reference to material point analyses and laboratory tests. Theimplications of these results are finally discussed with reference to a specific engineering application: the triggeringof landslides in unsaturated shallow slopes.

2. Stability theories for geomaterials: A brief review

From an engineering point of view, the assessment of failure involves the identification of limit domains inthe stress space (i.e., failure envelopes). Thresholds for the admissible stresses are an essential element of inelasticconstitutive theories (most notably of the mathematical theory of plasticity). This logic can be adapted to unsaturatedsoils, for which the characterization of failure involves the enhancement of the existing strength criteria throughsuction-dependent stress/strength measures [46, 47, 77].

Such traditional views, however, do not capture several aspects of the mechanics of failure in soils and rocks. Ingeomaterials, in fact, inelastic processes are initiated much earlier than the loss of strength capacity, and have beenobserved to promote unstable responses even within stress domains that would be usually considered to be safe [63].In the case of fluid-saturated soils, instability mechanisms can be promoted by the interaction between the solidskeleton and the pore fluid, thus originating the fluidization of natural deposits [11, 68, 102]. Geomaterial instabilitiesare also possible under constrained kinematic conditions, as in the case of the localized compaction observed underone-dimensional compression [7, 64]. The initiation of these modes of failure involves the dependence of thestrength properties on current state of stress, kinematic conditions and pre-failure deformations, thus making thedefinition of failure criteria less obvious than for other solids of engineering interest.

A thorough review of failure criteria in geomaterials goes beyond the objectives of the present paper. Nevertheless,such richness of failure mechanisms motivates the need to formulate methods that can go beyond the simple notionof stress threshold, and that can consider the role of static-kinematic conditions and incremental perturbations [23,63, 81]. Such tasks have been at the core of the scientific research on geomaterial stability, and have led to a varietyof concepts useful to interpret the instability modes observed in experiments and natural settings.

Within this context, the term “stability” has often taken different connotations when used in the frame of thetheory of plasticity. For instance, Drucker [42, 43] used the term to restrict the mathematical discussion to a specificclass of “stable” plastic solids. By contrast, Hill [59] made reference to the notion of stability while addressingthe uniqueness of the elastoplastic response of solids subjected to dead loads. In either cases, the use of the term“stability” was connected to a precise set of mathematical criteria, such as the loss of uniqueness and/or existenceof the plastic response. Since these pioneering works, a variety of methods have been proposed to identify loss ofuniqueness in mechanical analyses and discuss its relation with the physical notion of stability. In the domain ofgeomechanics, significant advances have been inspired by the further development of the concept of second-orderwork:

d2W = σ′ij εij (1)

Equation (1) is based on the standard use of the effective stress tensor,σ′ij = σij − uwδij , thus reflecting the validity

of the effective stress principle with respect to the stability of a fluid-saturated medium. This energy measure hasestablished a conventional criterion of stability [79]. Indeed, the positive definiteness of the quantity in (1) is widelyused to define the stability of the medium and the uniqueness of the incremental solution [59]. Considerable studiesbased on the use of d2W have recently led to the concepts of controllability [63] and sustainability [85], whichhave provided additional mathematical and physical meaning to the second-order work criterion. Indeed, while


Fig. 1. Saturated and unsaturated instability mechanisms: a) drained constant shear stress paths [data after 41]; b) volumetric expansion andcontraction in constant shear tests [data after 4]; c) stress paths for the wetting of unsaturated soil under a dead load [data after 29]; d) volumetricresponse of a test on an unsaturated soil specimen exhibiting fluidization instability [data after 90].

the concept of controllability provided a method for relating the occurrence of Hill’s criterion with the controlconditions imposed to a solid, the concept of sustainability explained the second-order work condition as the abrupttransition from static to dynamic deformation.

Although the connection of these ideas with the notion of stability in Lyapunov’s sense [78] is still an openquestion [27], the linkage between controllability, sustainability and bifurcation has favored the combined use ofmathematical methods and physical concepts in the description of the failure modes of soil specimens [85]. Aninteresting example is the case of constant shear drained tests in fluid-saturated sands, often used to model theeffects of rainfall events [4, 5, 25, 41, 83]. These experiments consist of two stages: drained shearing up to a givenvalue of deviatoric stress and a subsequent constant-shear path (during which the effective confinement is reducedby pressurizing the pore fluid). Numerous studies have shown that, even though these tests are initially conductedunder drained conditions, undrained failure can occur during the fluid pressurization stage because of an unstablemechanism (Fig. 1a). Such instabilities are generally observed to occur when the measured second-order work takes


negative values [41, 81, 83], which under constant deviatoric stress imply a transition from volumetric expansionto compaction (Fig. 1b).

This problem shares many analogies with the wetting stage of unsaturated soil samples subjected to a dead load(Fig. 1c). These saturation paths are usually imposed by a controlled increase of the pore water pressure [29, 90,103]. Also in this case, loose soils tend to undergo compaction upon suction-removal, with a phenomenology verysimilar to that in Fig. 1b. Nevertheless, catastrophic collapse and complete fluidization may or may not be observeddepending on the state of saturation and the properties of the soil [29, 90]. In addition, at variance with saturatedspecimens, the moment of collapse does not coincide with the point at which compaction begins. In contrast,collapses take place only at a later stage of the saturation process, when the degree of saturation is sufficientlyhigh for flow failure to take place (Fig. 1d). How is it then possible to reconcile the combined role of stress state,deformation patterns and hydraulic processes with the usual geomechanical interpretation of failure events? Thenext sections provide a vision to address this fundamental question in the framework of unsaturated soil mechanics.

3. Second-order work input to unsaturated soils

As previously discussed, the current understanding of instability in saturated soils benefits from the computationof the incremental energy input to the material, as well as from the relation of this measure with the loss ofcontrollability of the imposed loading parameters. The first crucial step to address the stability of an unsaturatedsoil is therefore the definition of the second-order energy input to an unsaturated soil volume. An enhanced definitionof the second-order work can indeed reconcile the stability of the unsaturated medium with the possibility to controlthe mechanical and hydrologic variables that govern its incremental response. The redefinition of a second-orderenergy input involves the identification of the perturbations acting at the boundary of an arbitrary volume ofunsaturated soil (Fig. 2). The incremental work done by the agents altering the state of the material can be expressedas follows [17]:∫

V

d2W · dV = 1

2

∫V

(Fwi w

wi + F ai w

ai + F si vi

) · dV + 1

2

∫A

(f wi w

wi + f ai w

ai + f si vi

) · dA (2)

in which F (−)j are the incremental body forces acting on each phase within the volume V , f (−)

j are the incremental

surface forces acting over the outer surfaceA,w(−)i are the velocities of the fluid phases and vi is the velocity of the

solid phase. Superscripts w, a and s stand for the water, air and solid phases, respectively. The integral in Equation(2) is performed over the entire unsaturated soil volume. As a result, the three phases are assumed to be smearedover the entire space (assumption of superposed continua). This is considered here by expressing the three termsf

(−)j through a notation typically employed in volume averaging theories [56]:

∆uw

∆uaOuter Surface, A

Fig. 2. Arbitrary volume schematic of an unsaturated geomaterial.


f wi = −nSruwδijnjf ai = −n (1 − Sr) uaδijnj

f si = − (1 − n) tij nj

(3)

where a compressive positive convention has been used. The term tij is the average stress acting over the solidgrains, uw and ua are pore water and pore air pressure, respectively, and nj is the outward normal to the surfaceA. This averaging procedure is based on the state of density (i.e., the current porosity n) and saturation (i.e., thecurrent value of degree of saturation, Sr).

By assuming an isothermal process, Equation (2) can be simplified by ignoring the terms inside the volumeintegral. As a result, once Equations (3) are introduced in (2), the following relation is obtained:∫

V

d2W · dV = −1

2

∫A

{n

[Sruwδijw

wi + (1 − Sr) uaδijw

ai

] + (1 − n) tijvi}nj · dA (4)

By assuming that the three continua act in parallel and that the stresses acting on each phase can be weightedwith the respective volume fraction, the following definition is obtained for the total stress:

σij = nSruwδij + n (1 − Sr) uaδij + (1 − n) tij (5)

Also in this case, Equation (5) is based on a volume averaging procedure. In addition, it is useful to introducethe seepage velocities, as follows:

wwi = nSr(wwi − vi

)wai = n (1 − Sr)

(wai − vi

)(6)

By considering (5) and (6) and using the Gauss theorem to convert the surface integral into a volume integral,the second-order work density is further simplified as follows:∫

V

d2W · dV = −1

2

∫V

{uwδijw

wi + uaδijw

ai + σijvi

},j

· dV (7)

Finally, by considering that the three phases are in equilibrium and by using the mass balance equations forair and fluid flow (see Buscarnera and di Prisco [17] for more details on this procedure) it can be shown that thesecond-order work input per unit volume reads as follows:

d2W = 1

2

[σij − Sruwδij − (1 − Sr) uaδij

]εij − 1

2n (ua − uw) Sr (8)

Equation (8) provides a generalized expression of the second-order work written in terms of local stress-strainvariables. This result is valid for an incremental loading/wetting process on an unsaturated porous medium, thusincluding the effect of both stress increments and pore-fluid fluctuations. The incremental stress measure conjugateto the strain rate in the second-order work equation is given by:

σ∗ij = σij − Sruwδij − (1 − Sr) uaδij (9)

Similarly, the incremental variable associated with changes in saturation conditions is defined by:

s∗ = n (ua − uw) (10)

The expression of d2W in Equation (8) indicates that the second-order work input in a partially saturated mediumdepends simultaneously on two components: a first one that is mechanical in nature (i.e., it is directly associatedwith the skeletal deformations) and a second one that depends exclusively on the evolution of the hydraulic state.

Equation (8) can be rearranged to obtain a useful representation of the second-order work in terms of thevariables commonly governed in laboratory tests. By adopting the strategy suggested by Houlsby [60], it is possibleto rearrange the terms appearing in (8), obtaining:


d2W = 1

2

[σij − uaδij

]εij − 1

2(ua − uw) ew/ (1 + e) (11)

Therefore, two equivalent possibilities for the second-order work can be written as follows:

d2W = 1

2σ∗ij εij − 1

2s∗Sr (12-a)

d2W = 1

2σnetij εij − 1

2sew/ (1 + e) (12-b)

where σnetij = σij − uaδij and s = ua − uw. Given the strategy used to derive (12), the above expressions do notinclude specific assumptions on the constitutive response of the medium. In addition, they are by no means the onlypossible options, as the components of the second-order work equation can be rearranged according to specificmodeling needs. Equations (12) can be therefore considered as the building blocks of incremental constitutiveequations for stability analyses. The essential difference with respect to the classical expression of d2W in Equation(1) is the ability to consider how the external energy input is distributed across the components that constitute anunsaturated soil (thus considering the effect of pressures and volume fractions of the pore fluids). This key featurederives from the presence of two main contributions: one associated with the skeleton and the other with the porefluids. It must be noted that Equations (11)-(12) have been obtained by using the conventional hypothesis thatsuction has isotropic effects on the skeleton stress. While recent findings suggest that this assumption may not holdunder all conditions [99], the vast majority of hydro-mechanical models for unsaturated soils do not include tensoreffects generated by suction. As a result, Equations (11)-(12) allow a straightforward inspection of the stabilityconditions in light of the predictions obtained from the most widely adopted constitutive models for unsaturatedsoils available in the literature.

4. A theory of hydro-mechanical controllability

4.1. Controllability of laboratory tests on unsaturated soil samples

Similar to Hill’s result [59], the extended expression of d2W in Equation (12) quantifies the second-order energyinput to an unsaturated soil volume. The latter is the outcome of hydro-mechanical alterations of the boundaryconditions of the considered domain. Energy considerations can therefore be used to show that non-positive valuesof d2W relate with the instability of the medium, since this circumstance is associated with accelerations and thepotential transition to a dynamic regime of deformation [84].

Such processes, however, must be framed in a hydro-mechanical constitutive framework. In other words, itis necessary to specify what hydro-mechanical paths promote instability and how such events relate with theconstitutive properties of a partially saturated soil [17]. Again, Equation (12) motivates a strategy to address theproblem. The extended expression of d2W , in fact, includes both mechanical and hydraulic terms [16]. As a result,it suggests that the most general form of constitutive relations associated with hydro-mechanical fluctuations canbe expressed as follows:

�A ={

�∗

s∗

}=

[Dmm

Dhm

Dmh

Dhh

]·{

ε

−Sr

}= DHM · EA (13)

where �A and EA are vectors collecting the hydro-mechanical stress-strain measures appearing in d2W , whileDHM is the tangent constitutive operator linking their increments. Equation (13) details the scenario in which d2W

is written as in (12-a). Other options, however, are possible. For instance, by considering (12-b), it is possible to state:

�B ={

�net

s

}=

[�mm

�hm

�mh

�hh

]·

⎧⎪⎨⎪⎩

ε

ew

1 + e

⎫⎪⎬⎪⎭ = �HM · EB (14)


in which a different set of constitutive variables is used for the two hydro-mechanical vectors �B and EB, thusredefining the constitutive matrix linking their increments, �HM.

The use of energy consistent incremental constitutive laws enables the reinterpretation of hydro-mechanicalinstabilities in the light of the Hill’s criterion. For example, the use of the coupled constitutive matrix DHM allowsone to relate the vanishing of the second-order work to some specific mathematical properties of the constitutiveoperator. Starting from (13), the second-order work can be expressed in a general form as follows:

d2W = 1

2�T · E = 1

2ET · DHM · E = 1

2ET · DS

HM · E (15)

where the superposed “T” stands for transposed. The onset of a generalized instability can be associated with theloss of positive definiteness of the matrix DHM in Equation (15), and then to the loss of positive definiteness ofits symmetric part DS

HM [17, 63]. The first point at which the positive definiteness of DSHM is lost coincides with

the first moment at which the Hill’s stability criterion can be violated, with hydro-mechanical unstable phenomenabeing possible for the very first time. It is then readily apparent that the non-symmetry of the constitutive matrix isdirectly associated with the occurrence of latent instabilities. In the case of unsaturated soils there are two possiblesources of non-symmetry: the non-associativity of the plastic flow rule (affecting the non-symmetry of term Dmmin Equation (13)) and the hydraulic degradation of the material (giving origin to term Dmh, which is in generaldifferent from Dhm). The presence of hydraulic degradation terms, in particular, represents an additional andindependent source of non-symmetry with respect to fully saturated conditions. Under particular circumstances,instabilities are therefore possible even in the case of associated flow rules. This produces an apparent paradox,since under particular conditions an unsaturated soil can be even more prone to develop instability than a saturatedsoil [17]. This paradox is explained by the presence of constitutive functions describing the hydraulic degradationeffects, which are therefore a crucial component for the prediction of coupled hydro-mechanical instabilities.

The relation between incremental perturbations and potential for failure can be extended to a hydro-mechanicalcontext by adapting the concept of controllability [63] to the case at stake. A simple example is the case of stress andsuction control, for which the control variables are already included in the vector, �B = { σnet s }T , the resultingresponse variables being given by, EB. From the theory of controllability it can be shown that in this case whenever,

det(�HM) = 0 (16)

the stress-controlled system no longer has a unique solution. By means of the controllability approach, it is possibleto also investigate more general situations involving mixed hydro-mechanical control conditions. Consider forinstance, the following set of control variables:

φ =

⎧⎪⎨⎪⎩

σnet

− ew

1 + e

⎫⎪⎬⎪⎭ (17)

in which the vector φ defines the loading programme through which the incremental loading path is imposed.Equation (17) encapsulates the control variables that are imposed during either water-undrained loading or water-injection under constant dead loads [18]. By using Equation (17), the hydro-mechanical incremental relations (14)can be rearranged as follows:

φ =

⎧⎪⎨⎪⎩

σnet

− ew

1 + e

⎫⎪⎬⎪⎭ =

[�mm −�mh�

−1hh�hm

−�−1hh�hm

�mh�−1hh

�−1hh

]·{ε

s

}= X · ψ (18)

in which it has been assumed det�hh /= 0 and X represents a coupled hydro-mechanical control matrix, while thevector ψ collects the response variables (i.e. the set of variables conjugate to φ in the second-order work equation).An instability in the sense of the controllability theory is predicted when either no solution or infinite solutions arepossible for a vanishing φ, i.e. when:


det (X) = 0 (19)

By virtue of the Schur’s theorem [100], it can be demonstrated that this condition coincides with:

det (�mm) = 0 (20)

This procedure shows how it is possible to determine the condition for loss of controllability corresponding tothe vanishing of the second-order work. Such a condition has been retrieved for the loading programme given byEquation (17), which can be interpreted as a generalized undrained failure condition for an unsaturated soil undertriaxial compression. Such a strategy can be applied to any form of hydro-mechanical incremental loading, to obtainmathematical conditions similar to Equation (20) that can be used to capture the onset of latent instabilities duringsaturation and locate the associated instability domains in the stress space [18].

4.2. Constitutive modeling of unsaturated soil response

The analysis of the stable/unstable response of unsaturated soils subjected to hydro-mechanical paths requiresthe link between the mathematical framework expounded in the previous sections and a predictive constitutivemodel. Any constitutive law for unsaturated soils can in principle be used for this purpose, and indeed, severaloptions have been proposed in the literature (see for instance the extensive reviews by Gens et al. [51], Gens [50]and Sheng [101] for a thorough account on constitutive modeling for unsaturated soils). Here, in order to provide aseries of examples valid for elastoplastic constitutive laws, the hydro-mechanical model developed by Buscarneraand Nova [15] will be used. This constitutive law enables the use of a non-associated flow rule and is formulatedby means of a modelling strategy tailored to reproduce the first-order features of unstable mechanisms in saturatedand unsaturated soils. These characteristics are guaranteed through constitutive functions that are defined in termsof the so-called average skeleton stress:

σ′′ij = σij − Sruwδij − (1 − Sr) uaδij (21)

Equation (21) automatically reproduces the increase in stiffness and shearing resistance due to unsaturatedconditions, as well as the onset of shear failure resulting from saturation processes. In addition, the stress measure(21) automatically reverts to the usual definition of effective stress for saturated soils when Sr = 1. The yieldsurface and the plastic potential of the selected model are characterized by the versatile expression proposed byLagioia et al. [71]. Furthermore, a full three-invariant dependence is incorporated in this expression by describingcritical state conditions through the Lode angle dependency suggested by Gudehus [54] and represented in Fig. 3.In order to reproduce the inelastic effects originated by wetting paths (a relevant example being the phenomenon ofwetting-induced compaction) the model incorporates a dependence of the yield locus on the hydraulic state variables.Different strategies have been proposed in the literature to incorporate such evolution of the yield surface (e.g., somemodels use suction-dependent elastic domains [1, 75, 88], while others incorporate the degree of saturation in thehardening rule [66, 101, 112]). Nevertheless, the analytical inspection of the mathematical structure of constitutivelaws for unsaturated soils suggests that the mathematical implications of the coupling between yielding domain andhydraulic state variables are very similar in all classes of unsaturated soil models [14]. As a result, the methodologyused in the following can be considered to be applicable regardless of the specific modeling choices done for thehydraulic variables appearing in the constitutive functions. For instance, in the model by Buscarnera and Nova[15] the evolution of the yield surface is reproduced by including a hydraulic effect in the hardening law driven bychanges in the degree of saturation, as follows:

p′′s = p′′

s

Bp

(εpv + ξsε

ps

) − rswp′′s Sr (22)

where p′′s is the internal variable defining the size of the yield surface, εpv and εps are the volumetric and deviatoric

plastic strains, respectively, while Bp, ξs and rsw are hardening parameters. The dependence of p′′s on the degree

of saturation reproduces the expansion/contraction of the yield surface upon drying/wetting processes (Fig. 4) andimplies the coupling between the retention properties (here reproduced through an uncoupled non-hysteretic van


Fig. 3. Yield surface representation in the deviatoric plane (a) and in the three-dimensional principal stress space (b) [16].

Fig. 4. Contraction of the yield surface with wetting in terms of mean skeleton stress (a) and mean net stress (b) [15].

Genuchten model [108]) and the mechanical response of the material. Simulations of one-dimensional compressiontests allow the implications of these constitutive assumptions to be described. Figure 5a shows two stress pathspredicted for saturated and unsaturated conditions, respectively. Numerical simulation of an increase in suctionprior to one-dimensional loading allows the effect of the parameter rsw to be disclosed. The expansion of the initialelastic domain postpones the onset of yielding and reduces the amount of predicted plastic strains upon loading. Ifwetting paths are eventually simulated, the yield surface shrinks in size, thus promoting plastic strains as a plasticcompensatory mechanism (Fig. 5b). It is possible to show that by increasing the values of rsw and Bp, the predictedamount of plastic compaction also increases, thus exacerbating the potential for hydro-mechanical instabilities[22]. The prediction of possible instability modes activated by such processes requires incremental constitutiveformulations able to accommodate the notion of material stability in accordance with Equation (12) [17].

4.3. Theoretical interpretation of model simulations

This section discusses a series of material point simulations characterized by constitutive instabilities. Thepurpose of the examples is to show how the theory of hydro-mechanical controllability can be used to inspect modelpredictions and detect unstable mechanisms. In particular, it will be shown how the theory can be used to identifyinstability modes that are not immediately apparent from the simulated paths, i.e. unstable processes that exist onlyin latent form, and whose activation is contingent on the presence of specific control conditions. These potentialmechanisms will be referred to as latent instabilities.

The examples are based on the simulated triaxial test results discussed in previous works by Buscarnera and Nova[15, 16]. The interpretation of the simulated tests is done by tracking the determinant of the constitutive matrix


0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80

Dev

iato

ric

Str

es

s, q

(k

Pa

)

Average Skeleton Stress, p'' (kPa)

Yield Surface (S

r=0.5)

Yield Surface (Sr=1)

Oedometric Path(S

r0=0.5)

a)

Oedometric Path(S

r=1.0)

0

0.02

0.04

0.06

0.08

0.1

0.1210 100

Vo

lum

etr

ic S

tra

in,

εε εε v (-)

Net Vertical Stress, σσσv

net (kPa)

Wetting-induced deformation

b)S

r0=0.5

Sr=1

Fig. 5. Oedometric stress paths for saturated and unsaturated conditions (a) and the corresponding volumetric response (b) [22].

associated with selected sets of control variables (i.e., associated with different options for the control vector φ).At the same time, the value of second-order work computed in light of the model is monitored, with the purpose ofproviding an energy interpretation of the constitutive singularities.

Each controllability scenario defines a specific expression of the constitutive control matrix and, hence, a spe-cific index of controllability, det (X). A loss of controllability can be caused by either mechanical or hydraulicperturbations. The first case considered here involves the application of deviatoric stresses under water-contentcontrol. This case is represented by the control vector in Equation (17), and is particularly interesting in the caseof water-undrained shearing. In the latter circumstance, in fact, the application of external stresses is accompaniedby changes in suction and degree of saturation. As a result, these paths involve coupled effects, which modify thepredicted mechanical properties of the unsaturated medium at any step of the simulation. In other words, the sameamount of deviatoric loading may induce radically different effects depending on the initial degree of saturationprior to shearing, the plastic compressibility and the initial values of suction and confinement stresses.

Figure 6 provides an example of such predictions [16]. Four triaxial tests at constant water content are simulated,starting from the same level of net confinement but different saturation conditions (and, hence, different initialsuctions). With the only exception of test #4, all the simulated paths exhibit a peak in deviatoric stress prior to thefrictional failure locus. In the case of the saturated specimen (test #1) the peak coincides with the onset of staticliquefaction. Simulations #2 and #3 share many similarities with these unstable patterns. However, while in test #2the peak is predicted to occur as an outcome of shearing-induced saturation (i.e., the volumetric strains due to theprior loading phase saturate the simulated specimen, thereby promoting a response very similar to that in simulation#1), in test #3 the peak is met prior to full saturation. In other words, the predicted instability in simulation #3is an outcome of the simultaneous effect of shearing, reduction in suction and saturation-induced weakening ofthe medium. Such predicted response is remarkably affected by the initial degree of saturation. Indeed, a furtherdecrease of the initial value of Sr postpones the onset of inelastic compaction, allowing failure to occur on the usualfrictional strength envelope (test #4).

When failure takes place within the unsaturated regime, the interpretation of the instability processes can beassisted by the second-order work measure given in (12b). Indeed, under axisymmetric stress conditions Equation(12b) can be written as:

d2W = 1

2pnet εv + 1

2qεd − 1

2sew/ (1 + e) (23)


−200 0 200 400 600 800 10000

200

400

600

800

1000

Mean Skeleton Stress, p’’ [kPa]

Dev

iato

ric S

tres

s, q

[kP

a]CSL

Test #4Sr

0=0.6

Test #1Sr

0=1

Test #2Sr

0=0.9

Test #3Sr

0=0.7

0 0.5 1−5

0

5

10

15

20x 10

15

normalized calculation step

det(

X) 1

0 0.5 1−2

−1

0

1

2

x 1015


det(

X) 2

0 0.5 1

0

5

10

15

x 1013


det(

X) 3

0 0.5 1

0

0.5

1

1.5

2

2.5x 10

14


det(

X) 4

Test #1 Test #2

Test #3 Test #4

Elastic phase

Plastic phase

Full saturation

Plastic phase

Elastic phase

Plastic phase

Plastic phase

Elastic phase

a)

b)

Fig. 6. Constant water content triaxial test results for initially saturated and unsaturated conditions: a) stress paths; b) stability response asreflected by det(X) [16].

where pnet is the increment in isotropic net stress (pnet = p− ua), q the increment in deviatoric stress, s the suctionrate, e the void ratio and ew = eSr the so-called water ratio (i.e., a measure of the volume of water in the medium).During a water-undrained triaxial compression test only the axial stress is increased, while the radial stress is keptconstant. As a result, the loading programme imposes pnet = q/3. At instability, however, q = 0 and, therefore,also pnet = 0. This implies no mechanical contributions to the second-order work. From a hydraulic viewpoint, thetest imposes a water undrained constraint, according to which ew = 0. Therefore, also the last term in Equation (23)is equal to zero at instability, implying the vanishing of the newly defined hydro-mechanical second-order work. Itis worth noting, however, that by comparing the trend of d2W in test #3 and #4 (Fig. 7), this quantity vanishes intwo different manners. While a transition from positive to negative values of second-order work is predicted in test


Test #4

0

100

200

300

400

500

600

700

800

0 0.05 0.1 0.15 0.2 0.25 0.3

Deviatoric Strain, edev

(-)

Peak stress of the stress-strain curve

P

Dev

iato

ric

Str

ess,

q (

kPa)

a)

0

200

400

600

800

1000

0 0.05 0.1 0.15 0.2 0.25 0.3


(-)

Failure achievedonly at large strains

P

Dev

iato

ric

Str

ess,

q (

kPa)

b)

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Deg

ree

of

Sat

ura

tio

n,

Sr (

-)

Peak of the stress path

P


(-)

c)

0.5

0.6

0.7

0.8

0.9

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Deg

ree

of

Sat

ura

tio

n,

Sr (

-)

P


(-)

d)

0.05 0.1 0.15 0.2 0.25 0.3-1×10-4

-5×10-5

0

5×10-5

1×10-4

Vanishing of d 2W:possible onset of

hydro-mechanical instability

P

Sec

on

d-o

rder

wo

rk,

d2 W

(kN

m/m

3 )


(-)

e)

0.04 0.08 0.12 0.16 0.2 0.24 0.280

5×10-3

1×10-2

1.5×10-2

2×10-2

2.5×10-2

P

Sec

on

d-o

rder

wo

rk,

d2 W

(kN

m/m

3 )


(-)

f)

Test #3

Fig. 7. Evolution of deviatoric stress (a–b), degree of saturation (c–d) and second-order work (e–f) as a function of the deviatoric strain for Test#3 and Test #4 in Fig. 6. Point P indicates the onset of instability, as predicted by the second-order work [16].


Fig. 8. Stability of constant shear stress paths (a) using the criterion for suction control instability (c); stress path as conditions change to watercontent control at points P and R (b), which agrees with predictions from the stability index for water content control (d). Points O1 and O2

mark predicted failure, and points T and Q indicate when saturation is reached. Modified from Buscarnera and Nova [16].

#3, test #4 exhibits a smooth evolution of d2W , which asymptotically attains a vanishing value in correspondencewith the critical state locus.

Shearing paths, however, do not reflect the mechanical effects of naturally occurring saturation events, such asrainfall infiltration. In such circumstances, in fact, the hydrologic variables may evolve with negligible alterations ofthe external stresses. It is therefore interesting to explore how such paths may affect the state of stability and whetheror not the currently applied state of stress is sustainable under the imposed conditions. Such analysis is convenientlysimulated by imposing suction removal under constant net stresses (Fig. 8a). In this case, the control variables arethose collected in vector �B (14), thus implying that only the singularity of the entire constitutive matrix (16) resultsin a loss of controllability. Such analysis is shown in Fig. 8c, showing that only the test performed at a sufficientlylarge value of initial deviatoric stress can undergo failure under the imposed conditions. In other words, the processof suction removal deteriorates the shearing resources of the material and causes a classical shear failure.

While this case is the most common, it does not reflect the entire range of possible failure modes. It is thereforenecessary to consider other potential mechanisms, such as those associated with water-content control. Indeed, bytracking the index (20), potential instabilities are found even for tests wetted at lower deviatoric stresses (Fig. 8d).To demonstrate the consequences of these results, suction control is interrupted in two points of tests #1 and #2(in both cases, during the constant deviatoric stress segment). At this point, suction control is replaced with watercontent-control shearing. This variation in constraints causes the latent instability identified in Fig. 8d to be activated,


Fig. 9. Stress path (a), water ratio response and collapse behavior (b) from a constant deviatoric stress simulation, with onset of instability (pointP) computed using the second-order work (c) [16].

and the resulting simulation for test #2 experiences a sudden decrease in strength (Fig. 8b). Alternatively, since test#1 had not lost resistance against water content-controlled loading, it initially remains stable, despite the changein loading conditions. The unstable nature of simulated path #2 is further elucidated by inspecting the value ofsecond order work, which becomes negative during the path. The points of d2W = 0 and det(X) = 0 coincide, andthe onset of unstable response is predicted to occur at the peak of the water ratio, while the void ratio decreasesthroughout the simulation (Fig. 9). These results show remarkable similarity with undrained collapses in saturatedsoil specimens. At variance with saturated conditions, however, the theory indicates that the unstable states do notcoincide with the onset of compaction, but with a non-trivial condition associated with the peak of the evolvingwater content. This result provides an interpretation of the differences between volumetric collapse in saturated andunsaturated soils. In other words, the observed transition from stable to unstable conditions in partially saturatedmedia can be explained by making reference to an overarching theory. It is indeed only within the frame of theconsidered theory that it is possible to identify the correct variables to be used for capturing and/or predicting thepoints of potential instability.

5. Implications for the triggering of shallow landslides

5.1. Stability criteria for shallow slopes

The typical strategies to address stability problems in unsaturated soil slopes involve the combination of transientinfiltration analyses with frictional failure criteria specialized to unsaturated conditions [12, 44, 49, 107]. Recentstudies have tried to extend this view to more general failure mechanisms by framing the onset of failure inunsaturated slopes within the context of second-order work criteria [26, 72]. While these works pointed out theimportance of advanced stability principles in the prediction of hydrologic-induced failures, they did not considerexplicitly the possible second-order energy effects of fluctuations in the state of the pore fluids. At variance withthis strategy, here we discuss an approach for slope stability analyses that is based on the enhanced expression ofsecond-order work discussed in the previous sections, thus incorporating the crucial effect of evolving pore-fluidpressures and volume fractions into the stability operators. As expounded by Buscarnera and di Prisco [18], it ispossible to elaborate the theory of hydro-mechanical controllability and apply it to slope stability analyses. In thismanner, the concept of controllability can assist the interpretation of the causes that promote the conversion of slopefailures into flows. Such transitions are usually attributed to liquefaction processes [68], which are schematicallyillustrated in Fig. 10a, c for a saturated infinite slope. Depending on the shearing scenario (either undrained ordrained), different failure modes can take place. In liquefiable deposits, in particular, the shear perturbations leadingto liquefaction (�τliq) are significantly smaller than those associated with drained failures (�τsf ), thus implyingthat soil liquefaction might represent the most critical failure scenario.

In unsaturated contexts, rainfall implies fluctuations of the in situ water pressure regime. Thus, the distance fromfailure conditions can be defined in terms of changes in suction, �s, i.e. the external forcing actively altering the


Fig. 10. Schematics of shear perturbation acting on a submerged infinite slope (a) and rain infiltration on an unsatured infinite slope (b); potentialfailure modes from shear perturbation: shear failure, �τsf and static liquefaction, �τliq (c); potential failure modes from rain infiltration: shearfailure upon suction removal, �ssf and wetting collapse, �swc (d) [22].

state of the slope. The key issue is whether shear failure (�ssf ) can be anticipated by other forms of collapseinitiated by wetting (e.g.,�swc in Fig. 10b, d). This consideration allows one to reinterpret the engineering problempictured in Fig. 10a, c (saturated conditions) for the case of unsaturated slopes (Fig. 10b, d) by including suctionand degree of saturation as additional variables in slope stability analyses [18, 22]. In other words, the key challengeis to define the states at which unsaturated soil slopes achieve potentially unstable conditions that favor the onsetof solid-to-fluid transitions.

It is convenient to illustrate this procedure by making reference to a submerged infinite slope (Fig. 11). Thekinematics that follows from this geometric assumption can be defined on the basis of the reference system inFig. 11. In particular, plane strain conditions yield εχ = γηχ = γχξ = 0, while symmetry along the slope impliesεη = 0. As a result, the incremental mechanical response of any point within the slope can be expressed as:

[σ′ξ

τξη

]=

[D11 D12

D21 D22

] [εξ

γξη

](24)

where Equation (24) reproduces a simple shear kinematics. The direct terms (D11 and D22) provide the stressincrements due to a change in their work-conjugate strain variables, while the off-diagonal terms are necessaryto reproduce shear-induced pore pressure (D12) and changes in shear stresses with volumetric strains (D21),respectively.


Fig. 11. Diagrams of saturated infinite slopes with failure boundary conditions for drained (a) and undrained (b) shearing [18].

Fig. 12. Diagrams of unsaturated infinite slopes with failure boundary conditions for constant suction (a) and constant water content (b) conditions[18].

Fig. 13. Typical undrained shearing results for contractive and dilative soils, with undrained instability predicted at the shear stress peak for thecontractive soil (a); prediction of latent instability for water content control prior to shear failure, during a wetting process (b) [18].


Any assumption on the control conditions in (24) will have an impact in the predicted slope stability scenario. Inparticular, it is possible to devise two extreme cases for control conditions: (i) drained shearing and (ii) undrainedshearing. A sketch of these two cases is given in Fig. 11. While the loss of controllability associated with drainedshearing is directly reflected by the singularity of the constitutive matrix in (24) (which can be shown to occur whenD11D22 −D12D21 = 0; [18]), in the case of an undrained simple shear path (24) would be rearranged as follows:[

εξ

τξη

]=

[D−1

11 −D−111 D12

D21D−111 D22 −D21D

−111 D12

] [σξ

γξη

](25)

resulting in the following condition at the loss of controllability:

det(XL) = D22 = 0 (26)

Equation (26) can be used to define the loss of stability of the selected controls (Fig. 13a), where det(XL) isthe index for liquefaction potential. It can be readily shown that, while the stress paths of contractive soils exhibita peak shear stress when the instability index det(XL) vanishes, dilative soils tend not to fail through undrainedshearing mechanisms, but they rather exhibit drained localized failures.

The use of incremental stress-strain measures in accordance with (12) enables one to derive failure criteria alsofor unsaturated conditions. For this purpose, let us consider an unsaturated infinite slope. Equation (24) can beadapted to include unsaturated conditions:⎡

⎢⎢⎣σ∗ξ

τξη

ns

⎤⎥⎥⎦ =

⎡⎢⎣D11 D12 D13

D21 D22 D23

D31 D32 D33

⎤⎥⎦

⎡⎢⎢⎣εξ

γξη

−Sr

⎤⎥⎥⎦ (27)

where terms [D13,D23] and [D31,D32] represent hydro-mechanical coupling terms. In particular, [D13,D23] modelthe mechanical effects of a change in saturation conditions. Positive values for these terms allow one to reproducea hydraulic relaxation due to wetting paths (collapsing materials). Terms [D31,D32] describe instead the effect ofskeleton deformation on the retention capabilities. For the sake of simplicity, hereafter only the first coupling effectwill be considered, assuming D31 = D32 = 0. Nevertheless, the extension of the analysis to more general cases isin principle straightforward [14, 22].

Also in the case of unsaturated conditions, it is possible to distinguish two simple perturbation scenarios: (i)suction-constant shearing and (ii) water-undrained shearing (Fig. 12). These two scenarios can again be translatedinto suitable stability criteria. Under constant suction, the hydro-mechanical perturbation is purely stress controlled(i.e., the disturbance to the system is represented by the stress vector on the left hand side of (27)). As a consequence,suction constant failure takes place when:

det(XS) = D33 (D11D22 −D12D21) = 0 (28)

where use of D31 = D32 = 0 has been done, and where XS represents the appropriate matrix for suction controlconditions. Indeed, excluding instabilities in the retention behaviour, it can be shown that condition (28) coincideswith the onset of shear strain localization in a saturated/dry material, as described by the singularity of the acoustictensor [18, 40, 95]. In other words, if coupling effects on the retention behaviour are excluded, shear failure inunsaturated soils turns out to be governed by the same mathematical condition found for saturated and dry soils. Atthis reference, it is worth noting that the coupling between deformation and retention properties is a very commonfeature of unsaturated soils [48, 80, 94, 105]. As a result, coupled effects can play an important role also in thebifurcation conditions of plasticity models for unsaturated soils [14]. Nevertheless, it is important to remark thatthe strategy of analysis illustrated so far is not affected by specific constitutive choices for the retention response,thus allowing a straightforward adaptation of the stability indices in the case of fully coupled retention behaviour.

Water-undrained shearing can be reproduced via (27) by incorporating different control conditions. Similar toundrained shearing in saturated soils, the changes in pore-fluid pressures are a product of the stress-strain response,


since neither σ∗ξ nor ns are assigned. Considering that total stress increments derive from equilibrium equations, it

follows that:

[σξ

τξη

]=

⎡⎢⎢⎣D11 −

(D11 −D33

Sr

n

)Sr

nD12

D21 −D23Sr

nD22

⎤⎥⎥⎦

[εξ

γξη

](29)

where the condition σ∗ξ = σξ + sSr has been used and the water-undrained constraint has been imposed as:

Sr = Sr

nεξ (30)

On the basis of constraint (30), the changes in volume resulting from a shearing processes induce changes in thedegree of saturation. It follows that, whenever deformational processes undergo instability, an abrupt variation insaturation conditions can be predicted. The instability condition under water-undrained perturbations is obtainedby setting the determinant of the control matrix in (29) to zero, obtaining:

det(XW) = D∗11D22 −D12D

∗21 = 0 (31)

where:

D∗11 = D11 −

(D13 −D33

Sr

n

)Sr

n(32-a)

D∗21 = D21 −D23

Sr

n(32-b)

in which XW represents the constitutive matrix for water content control conditions.Besides the diagonal terms (32), the failure criterion (31) shares many similarities with the usual localization

condition. The differences with respect to a suction-constant scenario derive in fact from hydro-mechanical coupling,and are relevant only when termsD13 andD23 are non-negligible [18]. For instance, an intuitive case is representedby dry conditions (Sr = 0), at whichD∗

11 = D11,D∗21 = D21 and (31) converges to the usual localization condition.

By contrast, coupling terms can play a crucial role in unsaturated collapsible materials, for whichD13 andD23 arepositive. Depending on the values taken by D13 and D23, condition (31) can anticipate shear failure (here givenby condition (28)) and can be originated when the critical state is not yet reached (Fig. 13b). According to thisscenario, the predicted effects of rain infiltration and/or shearing processes can be dramatic: coupled instabilitiescan in fact be activated when (i) the soil has a residual potential for volumetric collapse and (ii) the material is notyet saturated, causing in this way an abrupt saturation of the pores [16].

5.2. Model prediction of the triggering stage and implications for landslide forecasting

The stability of unsaturated deposits during rainfall events can be studied by simulating the response of the slopeto wetting paths. Although the quantification of suction perturbations over time would require data from transientinfiltration analyses, it is possible to simplify the description of these effects by representing their perturbations assuction removal processes. In this way the changes in suction reflect the disturbances altering the current state ofthe slope during a rainfall event.

The material point simulations illustrated in the previous section can be used to condense the effect of materialparameters and slope characteristics (e.g., deposit thickness, slope inclination, etc.). In other words, the perturbationsable to induce an instability can be identified through the stability indices obtained from material point analyses.This implies the assumption that any point within the slope responds according to a simple shear deformation mode,thereby simulating the stress-strain paths during rain infiltration in terms of the control variables discussed in theprevious section. Such simulations can be used to define instability scenarios for given sets of initial saturationconditions, slope inclinations and types of disturbance.


Fig. 14. Effect of slope inclination on instability: stress paths (a) and constant suction instability index (b); effect of material compressibility(Bp) on instability: stress path with marked point of instability (c) and instability index for water content control (d) [22].

Figure 14 shows a series of simulations consisting of a deposition stage at given values of suction and slopeinclination and a subsequent saturation stage imposed under constant total stresses. Figure 14a illustrates the effect ofvarying slope inclination, α, and its significant impact on the prediction of shearing instabilities. The simulation withthe lowest angle of inclination (α = 25◦) experiences no instability over the prescribed loading, and the normalizedstability index for suction control (i.e., (28) normalized by the absolute value of the index computed for the initialfield conditions) remains positive as the material reaches complete saturation (Fig. 14b). However, for a slightlyhigher inclination (α = 33◦), the shear failure locus is approached, with instability occurring as full saturation isachieved. The steepest slope considered (α = 40◦) requires even less suction removal to reach the failure locus,and consequently, instability is achieved while the material is still unsaturated. These results confirm the intuitivenotion that slope inclinations have a significant impact on shear failure, making rainfall-induced instability possibleeven at relatively low saturation levels (Sr ≈ 62 %, Fig. 14b).


Fig. 15. Suction-controlled constant shear simulations with control conditions changed to constant water content loading (at point P) for differentslope inclinations, α: (a) change of control after saturated instabilty index (det(XL)) vanishes; (b) change of control before det(XL) vanishes[22].

Although suction-control is the most common hypothesis for simulating saturation paths, it was previouslydemonstrated that water content-controlled conditions can represent a more critical scenario, especially whensolid-to-fluid transitions are of concern. Therefore, the concept of latent instability motivates the inspection of thenormalized stability index for water content control (31). This will allow us to evaluate whether during any of thepaths illustrated in Fig. 14a, the slope is predicted to be susceptible to coupled hydro-mechanical failure modes.For this purpose, the stress path of a suction-controlled simple shear simulation at inclination α = 30◦ is presentedin Fig. 14c. As expected from the previous study on slope inclination, the simulation does not reach shear failure.However, the evaluation of (31) for different values of material compressibility (Bp) discloses an increase of thepotential for latent instability. Indeed, the model constantBp reflects the potential of the material to exhibit inelasticcompaction upon wetting. Increasing values of this property cause a rapid decrease of the stability index (31),and hence latent instabilities can be identified before full saturation is reached (Fig. 14d). These analyses suggestthat a marked potential for volumetric collapse extends the range of slope angles susceptible to instabilities. Thisprediction has two critical implications: (i) even slope angles usually considered to be stable can be prone to generatelandslides, and (ii) the potential triggering mechanism is not always based on a frictional failure mode, but it mayinvolve the loss of control of the hydraulic parameters (e.g., suction and degree of saturation) in a similar mannerto what is observed during liquefaction events in saturated soil slopes.

The modes of loss of controllability predicted in Fig. 14 are contingent to specific control conditions. In otherwords, in order to be activated they require a convenient change of control. Nevertheless, their predicted occurrencecoincides with vanishing values of the second-order work, thus reflecting a potential for a spontaneous instability[85]. If a loss of stability of this kind produces a sharp saturation of the medium, it is necessary to inspect the responseof the material after such instability has taken place. The consequences of an unstable saturation can be furtherinvestigated by imposing appropriate changes in control conditions. An example of this analysis is given in Fig. 15,in which constant water content paths are imposed after the index det(XW) reduces below zero. Two differentslope inclinations are considered. In both instances, an immediate decrease in strength is predicted as the controlconditions change. For the steeper slope this transition occurs after the stability condition for saturated undrainedloading (det(XL)) has already vanished. Consequently, even after the simulation reaches saturation conditions, theunstable response continues through the activation of static liquefaction (Fig. 15a). In the second simulation, whichwas run at a gentle slope angle of α = 15◦, full saturation is reached when the material has not yet lost resistanceagainst saturated undrained shearing (Fig. 15b). As a result, upon saturation the material reaches a metastable


Fig. 16. Stability charts: suction control (det(XS)) and water content control (det(XW)) instability for water-insensitive soils (a) and soilswith hydro-mechanical dependence (b); effect of the saturated instabilty index (det(XL)) on the occurrence of immediate liquefaction (c) andmetastability (d). |�s|/s0 = 1 signifies full saturation [22].

state, which can withstand additional shear loading, until the index of undrained stability of the saturated medium(det(Xs)) also vanishes. Such predicted metastability has been disclosed by Buscarnera and di Prisco [22] and hasan impact in the quantification of the consequences of saturation-induced collapses. In other words, the notion ofmetastability can allow one to distinguish when saturation-induced collapses are actually able to turn into a flow.

The outcome of material point analyses can eventually be collected in graphical charts of triggering perturba-tions, here referred to as stability charts. These charts evaluate the normalized suction perturbation able to causeinstabilities (�sN = |�s| /s0) against the angle of slope inclination for different categories of material behavior.Any point of these charts is associated with the vanishing of at least one of the indices previously discussed. Thevertical axis indicates the magnitude of the normalized perturbation necessary to cause an instability, in whichcondition �sN = 1 is associated with the achievement of saturated conditions (i.e., |�s| = s0).

The different scenarios deriving from material point simulations are condensed in the schematic plots in Fig. 16.A first potential scenario applies to soils that are relatively insensitive to hydrologic perturbation, i.e., that do notexhibit significant compaction upon saturation (Fig. 16a). In this case, past a critical slope inclination (i.e., in zone2), the indices for suction control and water content control provide the same measure of triggering perturbation. Inother words, there is no distinction between the two modes of instability, and the slopes can fail only in a localizedshear failure mode that does not involve sharp changes in the hydraulic variables. By contrast, a different scenarioapplies to soils that display remarkable compaction upon wetting, i.e. that are characterized by considerable hydro-mechanical coupling (Fig. 16b). In this case, an additional chart can be found, which is associated with latentinstability and anticipates the chart of suction-controlled shear failure. A critical range of slope inclinations istherefore predicted, in which solid-to-fluid transitions are possible prior to shear failure and can be triggered by thesuction-removal path (zone 2a in Fig. 16b).

Some crucial characteristics of the instabilities promoted by saturation paths can be inferred by studying thestress-strain response following the unstable transition from unsaturated to saturated regimes (Fig. 15). This is


possible by comparing the chart associated with the potential for liquefaction of the saturated material (i.e., the chartassociated with det(XL) = 0) and the condition of saturation-induced collapse (associated with det(XW) = 0).In particular, if the condition for liquefaction of the saturated slope is always located below the latent instabilitychart of an unsaturated slope (Fig. 16c), the volumetric collapse caused by saturation is predicted to be followedby catastrophic liquefaction. By contrast, if for some slope angles the threshold for liquefaction failure has not yetbeen crossed when the latent instability becomes possible (zone 2a in Fig. 16d), the processes of suction-removalacting at those slope inclinations can induce metastable responses, i.e. a slope failure mechanism which is featuredby a transient state of instability that does not evolve in a complete fluidization.

The examples presented above highlight the effect of different factors (i.e., slope inclination, compressibility,solid-fluid couplings, etc.) and the consequences of hydro-mechanical control conditions. According to the predictedscenarios, solid-to-fluid transitions can be triggered by rainfall events only in particular cases, i.e., when the soilsare both liquefiable and prone to compact upon saturation. In such cases, flow failure is predicted to take place foran intermediate range of slope angles, which separates fully stable states from localized frictional slips. As a result,according to the theory the transition from solid to fluid response can be explained as a chain process, consistingof suction-removal, a sharp volumetric collapse causing saturation of the medium and, eventually, liquefactioninstability in the saturated regime.

6. Conclusions

This paper has reviewed a class of methodologies to detect bifurcation and failure in unsaturated soils. Althoughthis is a well-established area of research in dry and fully saturated geomaterials, the analytical frameworks for three-phase porous media can be considered to be still in their infancy. Indeed, even if numerous and wide-ranging advancesin experimental, constitutive and numerical modelling of unsaturated soils have already been achieved, the theoreticalmethodologies to detect instabilities, define their deformation patterns and capture the role of hydro-mechanicalcoupling and drainage conditions have attracted systematic attention only over the last decade.

The paper has focused on a set of methodologies recently developed by the first author and several other collab-orators. These endeavors have been aimed at providing a systematic organization of the main concepts related withinterpretation and prediction of failure mechanisms in unsaturated soils, with the main goal to provide a referencetheory conceptually similar to those already available for saturated soils. Reviewing the recent developments, it ispossible to identify a number of significant features that characterize unsaturated soil stability:

• The second-order work input is characterized by two terms: one of mechanical origin (which includes theincrement of a skeleton stress measure work conjugate to the strains) and one of hydraulic origin (i.e., associatedwith changes in suction and degree of saturation). The combination of these two contributions defines the sign ofthe incremental energy input, implying that both terms have a similar impact on the stability of the medium. Thisresult was derived in light of general premises, that did not involve assumptions on the constitutive behaviourof the unsaturated material nor on the effective stress measure that governs the stress-strain response of the soil.

• The generalized expression of second-order work motivates the incremental formulation of hydro-mechanicalconstitutive laws for material stability analyses. This procedure allows one to extend the notion of controlla-bility to hydro-mechanical processes, thus including the effect of hydrologic perturbations. Model simulationsbased on the framework of strain-hardening plasticity have been used to discuss the importance of specific soilproperties, such as those that govern the coupling between plastic yielding and water-retention behaviour. Theexamples show that the combined use of second-order work and controllability indices discloses unexpectedpotential instabilities caused by hydro-mechanical loading paths.

• When the theory is applied to the mechanics of failure in unsaturated shallow slopes, it is possible toelucidate the role of the controlling factors for landslide hazards, such as slope inclination, thickness of thedeposit, state of saturation and hydro-mechanical perturbing agents. The theory supports the identification ofmultiple stability indices and enables one to differentiate frictional slips and flow failures. The mathematicalinspection of model simulations provides a mechanistic interpretation for the solid-to-fluid transition causedby saturation processes, which can be described as a chain process, consisting of volumetric collapse, suddensaturation and catastrophic flow.


While the mechanical problems discussed in this review are still in large part the subject of ongoing research andcannot be considered to be conclusive, these ideas set a general vision to extend the bifurcation theory to the analysisof multiphase porous media, thereby using it as a predictive tool to evaluate the potential for failure processes inearthen systems. Towards this goal, further coordinated efforts on experimental, theoretical and numerical methodsare necessary, and can pave the way to answer numerous open questions in unsaturated soil mechanics. Physicallybased constitutive models need to support this endeavor, making it possible to cope with instabilities across theentire regime of saturation and elucidating the factors that govern localized and diffuse modes of deformation inunsaturated soils. In a future perspective, such methodologies also have the potential to be applied to the solutionof large scale geo-engineering problems. An example discussed in this review is the case of landslide hazards, forwhich synergies between advanced geomechanical methods, field investigations and remote sensing can support theidentification of the areas prone to generate catastrophic slope failures, thus contributing to mitigate the associatedhazards for human communities.

Acknowledgements

This work has been partially supported by the US National Science Foundation under Grant No. CMMI-1234031,Geomechanics and Geomaterials Program.

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