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

Transcription

1 Buscarnera, G. & di Prisco, C. Géotechnique [ Soil stability and flow slides in unsaturated shallow slopes: can saturation events trigger liquefaction processes? G. BUSCARNERA and C. DI PRISCO This paper illustrates an application of the theory of material stability to the analysis of unsaturated slopes. The main goal is to contribute to the understanding of rainfall-induced flow slides. For this purpose, a coupled hydromechanical constitutive model is combined with a simplified approach for the analysis of infinite slopes. Simple shear-test simulations are used to evaluate triggering perturbations and investigate the role of both initial suction and stress anisotropy in the activation of slope failures. The numerical simulations clearly show that different mechanisms of activation can be originated. The onset of instability is detected by introducing appropriate stability indices for distinct modes of failure: localised shear failure, static liquefaction and wetting-induced collapse. Critical intervals of slope inclinations are identified, cautioning that the predicted failure mode may change dramatically depending on initial conditions, slope angle and material properties. The numerical simulations demonstrate that, in particular circumstances, saturation of the pore space can be the unexpected result of a volumetric instability. According to this interpretation, a rainfall-induced flow slide can originate from a complex chain process consisting of a sudden volume collapse, uncontrolled saturation of the pores and, eventually, catastrophic liquefaction of the deposit. KEYWORDS: constitutive relations; landslides; liquefaction; partial saturation; suction; theoretical analysis INTRODUCTION In many parts of the world, geohazards pose serious threats to territory, economy and human lives. During recent decades, catastrophic events have been exacerbated by unpredicted climate changes and uncontrolled human activities (Cascini, ). The environment tends to be exposed to phenomena never experienced before, which now represent systematic causes of massive economic loss. Within this context, rapid landslides induced by rainfall represent a critical issue. These catastrophic events are characterised by rapid and unexpected activation, and are capable of mobilising huge volumes of material over large areas (Chu et al., 3; Olivares & Picarelli, 3; Picarelli et al., 8). The compelling need to capture the physical causes of such dramatic landslides requires a deep understanding of the phenomena involved, and advanced modelling strategies. This paper focuses on the study of landslides triggered by rainfall events, with the aim of investigating the mechanics of these processes and modelling their activation. Particular emphasis is given to the study of those landslides in which the soil suffers a phase transition from solid to fluid (here referred to as flow slides ). Such transition is usually attributed to a liquefaction process (Castro, 1969; Lade, 199), which is schematically illustrated in Fig. 1. Depending on the shearing scenario (either undrained or drained), different failure modes can take place, given that in a liquefiable deposit the shear perturbations leading to liquefaction ( ô liq ) are significantly lower than those associated with drained failures ( ô sf ). The main engineering implication of such a variety of instability modes is the existence of z Soil Bedrock Shear stress, τ Shear perturbation ( Δτ) σ n Shear failure locus τ α Δτ liq In situ stress Δτ sf Manuscript received 9 July 11; revised manuscript accepted 7 December 1. Discussion on this paper is welcomed by the editor. Department of Civil and Environmental Engineering, Northwestern University, Evanston, USA. Department of Structural Engineering, Politecnico di Milano, Milan, Italy. Normal effective stress, σ n Fig. 1. Schematic representation of a shear perturbation ô acting over a submerged infinite slope (ó9 n is the in situ normal effective stress); possible failure modes: difference in external perturbations needed to activate either drained shear failure ( ô sf ) or static liquefaction ( ô liq ) 1

2 BUSCARNERA AND DI PRISCO a multiplicity of safety factors. These considerations inspired the development of a wide spectrum of theories aimed at differentiating liquefaction from shear failure and evaluating the risk of flow slides in subaqueous sandy slopes (Poulos et al., 198; Sladen et al., 198; di Prisco et al., 199). In the reminder of the paper this logic will be exported to the case of sub-aerial slopes, for which a comprehensive framework of interpretation is not yet available. The purpose is to provide a consistent geomechanical explanation of failure processes in unsaturated deposits by answering three major questions. What is the amount of suction removal at which a slide can take place? How and when can slope failures evolve into a flow-like mass movement? (c) Are fully saturated conditions necessary to induce liquefaction, or can such collapses be initiated by the prior wetting process? To address such problems, a novel methodology has been developed, which is based on three cornerstones: the extension of the concepts of material stability to unsaturated soils (Buscarnera & Nova, 11; Buscarnera & di Prisco, 1) their application to the simplified scheme of infinite slope (di Prisco et al., 199; Buscarnera & di Prisco, 11a, 11b; Buscarnera & Whittle, 1) (c) the use of a coupled hydromechanical constitutive model for unsaturated soils (Buscarnera & Nova, 9). The paper is thus aimed at applying the theory already discussed in Buscarnera & di Prisco (11b) by employing a suitable constitutive relationship. The main goal of the investigation is to elucidate the mechanical processes involved in the triggering of flow slides in partially saturated soil slopes. THEORETICAL BACKGROUND Second-order work and controllability for unsaturated soils A generally accepted approach for identifying unstable conditions in solids is Hill s criterion (Hill, 198), according to which a sufficient condition for stability is the positive definiteness of the second-order work, d W. This criterion provides a physical interpretation for instability, given that negative values for d W can correspond to a spontaneous burst in kinetic energy (Sibille et al., 7), and can be used for studying the initiation of slope instabilities (Lignon et al., 9). In unsaturated contexts, a critical task is to incorporate the mechanical implications of saturation processes. In fact, changes in the degree of saturation imply additional energy contributions (Houlsby, 1997; Gray et al., 1), and require the adaptation of second-order work measures. A strategy for this extension has been recently suggested by Buscarnera & di Prisco (1), who showed that under unsaturated conditions d W becomes d W ¼ 1 _ó ij S r _u w ä ij ð1 S r Þ_u a ä ij _åij 1 n ð _u a _u w Þ S _ r also possible to identify incremental stress variables for second-order work analyses, as for instance and _ó ij ¼ _ó ij S r _u w ä ij ð1 S r Þ_u a ä ij _s ¼ n_s ¼ n ð_u a _u w Þ The extended expression for d W enables instability conditions for unsaturated geomaterials to be identified and linked to the mathematical concept of controllability (Nova, 1994; Imposimato & Nova, 1998; Buscarnera & Nova, 11; Buscarnera et al., 11). In order to describe this concept, consider a set of incremental hydromechanical constitutive relations linking the control variables _ö (i.e. the disturbance applied to the material) and the response variables _ł (i.e. the outcome of the response of the material), as _ö ¼ X _ ł () where X is the control matrix. If the hydromechanical variables in equation () are selected in accordance with equation (1), the loss of uniqueness and/or existence of the incremental response (i.e. det X ¼ ) corresponds to a vanishing second-order work (i.e. it violates Hill s criterion). The concept of controllability provides a further insight into the physical meaning of equation (1) defines the failure mode mathematically (through the eigenvectors of X) (c) permits an intuitive definition of instability that encompasses saturated (Imposimato & Nova, 1998) and unsaturated conditions (Buscarnera & Nova, 11). As will be expounded later, the most notable feature of this theory 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., 199; Buscarnera & Whittle, 13). Using second-order work principles in unsaturated infinite slopes The concepts of second-order work and controllability can be used to elucidate the hydromechanical properties that can play a role in the initiation of uncontrolled deformation processes. For this purpose, consider a reference system associated with an unsaturated infinite slope of a given inclination (Fig. ). By starting from the three-dimensional stress strain response of a material point (see the Appendix for such a representation), it is possible to represent the χ (out-of-plane coordinate) η ¼ 1 _ó ij _å ij 1 _s _ Sr (1) ξ Soil where ó ij is the total stress tensor, å ij is the strain tensor, ä ij is Kronecker s delta, _u w and _u a are the pore water and pore air pressure rates respectively, S r is the degree of saturation, and n is the porosity. For S r ¼ 1, the above expression converges to the usual definition of second-order work for saturated media (i.e. d W ¼ 1 _ó ij 9 _å ij, ó9 ij ¼ ó ij u w ä ij being the effective stress tensor). By rearranging equation (1) it is Bedrock Fig.. Reference system for unsaturated infinite slopes α

3 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 3 mechanical response of a point within the deposit as a simple shear deformation mode, having 8 _ó >< î >= D 11 D 14 D 17 >< _å î >= _Ó ¼ _ô îç >: _s >; ¼ 4 D 41 D 44 D 47 _ª îç D 71 D 74 D 77 >: S _ >; ¼ D E _ (3) r The two vectors Ó _ and E _ are linked by a coupled constitutive operator (matrix D) and collect the hydromechanical variables associated with the incremental energy input on an infinite slope. In particular, _ó î is the incremental skeleton stress along the direction normal to the slope, and _ô îç is the shear stress increment along the slope inclination, while _å î and _ª îç are their work-conjugate kinematic counterparts. The hydraulic variables associated with changes in saturation conditions (i.e. _s and S _ r ) are selected on the basis of equation (1). The loss of positive definiteness of d W is governed by the symmetry properties of D. By decomposing this matrix into the sum of a symmetric part, D s, and a skew-symmetric matrix, D sk, it can be shown that d W ¼ 1 _ Ó T _ E ¼ 1 _ E T D s _ E (4) and D 47 ), the effect of soil deformation on the retention curve is often negligible at shallow depths. It is worth noting that these observations have implications that are comparable to the use of a non-associated flow rule (i.e. they exacerbate the potential for instability), and may not apply to all classes of unsaturated geomaterials. For instance, if the water retention curve depends significantly on the void ratio (term D 71 6¼ ), compressive deformation can promote a shift of the retention curve towards higher suctions (possibly having beneficial effects in terms of stability). At this point it is worth noting that, although consistent advances have been produced in describing the effect of volumetric strains on retention capabilities (Romero & Vaunat, ; Gallipoli et al., 3), there is still little guidance for incorporating the effect of shear strains (term D 74 ). This fact complicates evaluation of the interplay between retention properties and soil stability. For these reasons, the effect of deformation on the retention curve will not be accounted for in the following developments, thus using a simpler modelling strategy, one that is consistent with the limited geomechanical evidence available for collapsible unsaturated soils involved in flow slides. given that Hill s criterion can be violated for the first time when det D s ¼ (Imposimato & Nova, 1998). It interesting to observe that, since det D s < det D (Ostrowski & Taussky, 191), possible non-symmetries of D imply that Hill s criterion can be violated before condition det D ¼ is satisfied. As a result, stress-suction control conditions may not be the most critical combination of control parameters underpinning the collapse of natural slopes, which thus suggests the existence of unexpected failure modes. For simple shear conditions this concept is exemplified by the skew-symmetric part, D sk, which embodies the difference between matrix D (whose singularity reflects suction-controlled failure) and its symmetric part D s (which reflects possible violations of Hill s criterion) D sk D ¼ 41 D D 71 D 17 D 14 D 41 D 74 D 47 3 D 17 D 71 D 47 D 74 7 Alternative modes of failure are promoted by the difference between terms [D 14, D 41 ] (reflecting non-associativity of the mechanical response) or between the pairs [D 17, D 71 ] and [D 74, D 47 ] (related to hydromechanical coupling). While [D 14, D 41 ] depend on the characteristics of the yield surface and the plastic potential, the other off-diagonal terms reflect different behavioural properties: terms D 17 and D 47 reproduce the inelastic effects of saturation paths (e.g. wettinginduced compaction), and terms D 71 and D 74 reproduce instead the dependence of the retention curve on volumetric and shear strains respectively. At variance with the effect of non-associativity on material instabilities, which has been widely studied for several decades (Rudnicki & Rice, 197; Bigoni & Hueckel, 1991; Lade, 199; Nova, 1994), the role of hydromechanical coupling still deserves special attention. Similar to the non-associativity of the plastic flow rule, hydraulic off-diagonal contributions must be assessed on the basis of experimental evidence. In this work, the assumptions for the hydromechanical contributions are motivated by the geomechanical characterisation of some unsaturated soils involved in recent flow slide events (Cascini & Sorbino, 4; Bilotta et al., ; Ferrari et al., 1). These studies suggest that, while suction effects can induce changes in the preconsolidation stress of the collapsible deposits (terms D 17 () APPLYING UNSATURATED SOIL MECHANICS TO STABILITY OF SUB-AERIAL SHALLOW SLOPES Stability indices for unsaturated shallow slopes By following the strategy proposed by di Prisco et al. (199), it is possible to derive stability indices in analytical form and use them for the stability analysis of shallow deposits. The extension of this procedure to unsaturated slopes has recently been expounded in Buscarnera & di Prisco (11b), and this paper is a numerical application of their analytical results. Hereafter, only some basic aspects of the theory are noted; the Appendix provides a description of the mathematical strategy used by the authors to derive the stability indices. Two triggering mechanisms are investigated: a translational slide taking place under constant suction (in this case permeability is assumed to be infinite); and slope collapse initiated under water-content control (e.g. water-undrained shearing, water inundation, etc.). These failure modes will be referred to as mode A and mode B respectively. Shear failure (mode A) can be considered the most usual form of material instability in slopes, and it is often the failure mechanism included in conventional stability analyses for unsaturated slopes (Ng & Shi, 1998; Gasmo et al., ). This mechanism is originated either by an increase in shear stresses or by a decrease in suction due to water infiltration. The former perturbation is conveniently represented by a change in stresses at constant suction, and the latter is often modelled through a decrease in suction at constant total stresses. In both cases the control variables coincide with those collected in the left-hand side of equation (3). By following Buscarnera & di Prisco (11b), a stability index for this mechanism can be defined as I AU ¼ D 11 D 44 D 14 D 41 (6) where the subscript U stands for unsaturated conditions. The above expression has been obtained by excluding singularities in the retention curve (i.e. D 77 6¼ ) and neglecting a possible role of strains in the retention behaviour (i.e. D 71 ¼ D 74 ¼ ). Under these assumptions, condition I AU ¼ coincides with the strain localisation criterion obtained by di Prisco et al. (199) for a saturated layer of an infinite slope, I AS ¼ I AU (7) where the subscript A refers to a shear failure mode, and S stands for saturated conditions.

4 4 BUSCARNERA AND DI PRISCO The second triggering mechanism (mode B) occurs when the water content is controlled. This mode is relevant either when water drainage is prevented by natural layering or when water inlets from a surrounding formation can be modelled as a fluid volume injected into the pores (Buscarnera & di Prisco, 11b). In either cases, changes in suction are no longer imposed, but are obtained as an outcome of the deformation of the porous medium. From a mathematical viewpoint this analysis is similar to passing from stresscontrolled to strain-controlled conditions, and influences the onset of bifurcation. The stability index associated with failure of the slope under constant water content has an expression that is very similar to shear failure, I BU ¼ D 11 D 44 D 14 D 41 (8) where the modified terms D 11 and D 41 reflect the role of hydromechanical coupling in the considered failure mechanisms (i.e. they also depend on degree of saturation, porosity and coupling terms D 77, D 17 and D 47 ). The derivations of D 11 and D 41, as well as their analytical expressions, are given in the Appendix. Buscarnera & di Prisco (11b) showed that when S r ¼ (i.e. when the role of solid fluid coupling vanishes), I BU coincides with I AU, and the two indices provide the same bifurcation mechanism. By contrast, when S r ¼ 1, the water-undrained deformation mode is naturally associated with the initiation of static liquefaction. It is possible to expound this conceptual link by deriving the hydromechanical control matrix associated with the control of total stresses and water content < _ó î = D 11 D 14 nd ð 77 Þ 1 D 17 S r < _å î = _ô : îç ; ¼ 6 D 41 D 44 nd ð 77 Þ D 47 _ª îç _e w = ð1 þ eþ S r nd ð 77 Þ 1 : ; _s (9) in which e is the void ratio and e w is the water ratio, defined as e w ¼ es r : Equation (9) can be used to reproduce the response of the slope when it is subjected to a hydromechanical perturbation. The cases of water-undrained shearing and water inundation under dead load are included as particular cases. It is possible to show that a singularity of the control matrix in equation (9) is governed by the stability index, equation (8) (i.e. its determinant vanishes when I BU ¼ ). When S r ¼ 1, simple physical considerations allow equation (9) to be rewritten to address the undrained loading of a fully saturated soil. In fact, under saturated conditions, changes in the degree of saturation are no longer possible and coupling effects disappear (i.e. D 1 77 ¼, thus giving nd 1 77 D 17 ¼ nd 1 77 D 47 ¼ ). In addition, since water is the only pore fluid, the incremental response can be expressed as < _ó î = D 11 D 14 1 < _å î = _ô : îç ; ¼ 4 D 41 D 44 _ª îç (1) : ; _e w = ð1 þ eþ 1 _u w in which the presence of the pore water pressure, u w, reflects the fact that the mechanical response is now governed by the effective stresses. The condition involving changes in water volume (i.e. _e w ) has the role of enforcing the fluid mass balance (thus imposing an isochoric kinematics, _å î ¼ ). It is straightforward to show that the control matrix in equation (1) vanishes when D 44 ¼ (11) Equation (11) coincides with the analytical condition for undrained failure under simple shear conditions (di Prisco et al., 199) and can be used to derive a stability index for static liquefaction, I BS ¼ D 44 (1) where the subscript B indicates an undrained failure mechanism under saturated conditions (subscript S). It is therefore shown that the bifurcation mode associated with I BU ¼ shares similarities with both shear strain localisation (indices I AS and I AU ) and static liquefaction (index I BS ). The effect of terms D 11 and D 41 implies that, in particular circumstances, mode B can occur before mode A. In other words, the hydromechanical constraint on the drainage of water implies that, depending on the soil properties, the features of this instability mode can be those of either a shear strain localisation or a liquefaction process (i.e. it can involve uncontrolled changes in pore water pressures). In the following, the particular features of these bifurcation modes will be elucidated by means of numerical simulations. In order to simplify the comparison among different initial conditions and failure modes, the instability indices presented in this section will be reported in a normalised form, as I ij ¼ I ij I ij (13) where I ij is a stability index and I ij is a positive reference value that, unless otherwise stated, will be the absolute value of the stability index for in situ conditions. A constitutive model for unsaturated soils The theoretical framework presented in the previous sections can in principle be combined with any constitutive law for unsaturated soils. Since the focus of this paper is to investigate the stability of unsaturated slopes, the hydromechanical model developed by Buscarnera & Nova (9) has been selected. This constitutive law permits the use of a non-associated flow rule, and is formulated by means of a modelling strategy tailored to reproduce first-order features of unstable mechanisms in both saturated and unsaturated soils. This is achieved through constitutive functions that are defined in terms of the so-called average skeleton stress ó ij ¼ ó ij S r u w ä ij ð1 S r Þu a ä ij (14) Equation (14) reproduces the increase in stiffness and shearing resistance due to unsaturated conditions, as well as the onset of shear failure resulting from saturation processes. Wetting paths, however, can originate inelastic effects much before failure (a relevant example being the phenomenon of wetting-induced compaction). A common strategy adopted to model these processes relies on the introduction of a dependence of the yield locus on suction and/or degree of saturation (Alonso et al., 199; Jommi & di Prisco, 1994; Wheeler et al., 3; Sheng, 11). This peculiar feature of unsaturated soils is reproduced here by including a hydraulic effect in the hardening law, as _p s ¼ p s _å p v B þ î s _å p s rsw p s Sr _ (1) p where p s is the internal variable defining the size of the yield surface, å p v and åp s are the volumetric and deviatoric plastic strains respectively, and B p, î s and r sw 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. 3), and implies coupling between the retention properties (here reproduced through an

5 Suction, s s 1 s s 3 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES Wetting path uncoupled non-hysteretic Van Genuchten model; Van Genuchten, 198) and the mechanical response of the material. Simulations of one-dimensional compression tests allow the implications of these constitutive assumptions to be described. Fig. 4 shows two stress paths, predicted for saturated and unsaturated conditions respectively. Numerical simulation of an increase in suction prior to one-dimensional loading allows the effect of the parameter r sw to be shown. The expansion of the initial elastic domain postpones the onset of yielding, and reduces the amount of predicted plastic strains upon loading. If wetting paths are eventually simulated, further plastic strains are predicted as a plastic compensatory mechanism initiated by suction removal (Fig. 4). Fig. illustrates the role of some of the material constants in equation (1), showing that larger values of r sw and B p are associated with a larger potential for wetting collapse. As was outlined in Buscarnera & di Prisco (1), the prediction of hydromechanical instability requires an incremental formulation able to accommodate the notion of material stability. For this reason the incremental constitutive equations of the model discussed in this section have been arranged in accordance with equation (1). This choice allows reinterpretation of the engineering problem pictured in Fig. S r1 S r S r3 8 Degree of saturation, S r 7 6 Suction, s s 1 s s 3 Deviatoric stress, q 4 3 Yield surface ( ) S r Oedometric stress path (unsaturated conditions) Oedometric stress path (saturated conditions) p s (saturated conditions) p s3 p s3 p s3 1 1 Initial state before drying ( S r 1) Average skeleton stress, p 7 8 Internal variable, p s ( s, S ) 1 r1 Unsaturated conditions Deviator stress, q ( s, S ) 3 r3 ( s, S ) r Wetting path Volumetric strain, ε v Saturated conditions Wetting-induced deformation 1 p s3 p s Mean skeleton stress, p (c) Fig. 3. Schematic description of retention curve and wetting path; hydraulic effects on evolution of internal variable p s ; (c) changes in size of elastic domain during wetting processes. Suction axes are to be considered in logarithmic scale p s1 1 1 Net vertical stress, σ net v 1 Fig. 4. Initial elastic domains and predicted stress paths for one-dimensional compression under saturated and unsaturated conditions (constitutive parameters given in Table 1); predicted volumetric response for two simulations and effect of saturation path imposed at constant vertical net stress

6 6 BUSCARNERA AND DI PRISCO Volumetric strain, ε v B p 38 B p 6 B p 9 Increase in plastic compressibility ( ) B p Soil Bedrock z ΔS α Water infiltration ( Δq w ) 1 14 r sw 3 87 Shear failure locus Volumetric strain, ε v B p 38 r sw 3 87 r sw 6 r sw Suction, s Increase in hydraulic hardening parameter ( ) r sw Shear stress, τ (* ss r ) Δs sf Δs wc Normal skeleton stress, σ n In situ stress before rainfall Fig. 6. Schematic representation of rainfall infiltration ( q w ) acting over unsaturated infinite slope and causing perturbation of suction at material point level ( s); possible failure modes: difference in suction removal needed to activate shear failure ( s sf ) or wetting collapse ( s wc ) Suction, s Fig.. Parametric analysis: effect of plastic compressibility (B p ) and hydraulic hardening parameter (r sw ) on the volumetric strains predicted upon suction removal 1 (saturated conditions) for the case of unsaturated slopes (Fig. 6). Rainfall events imply, in fact, a variation of the in situ water pressure regime. Thus the distance from failure conditions can be defined in terms of changes in suction, s: that is, the external perturbation altering the state of the slope. The key issue is whether shear failure ( s sf ) can be anticipated by other forms of collapse initiated by a wetting process (e.g. s wc in Fig. 6). In the following this logic will be used to evaluate the distance from instability conditions. Although such an incremental definition of failure does not coincide with the usual definition of safety factor, it has some advantages that are specific to the present analysis (and more in general to the evaluation of instability conditions from coupled elasto-plastic soil models), as it allows the same strategy to be used for general types of instability mode. Role of constitutive parameters in prediction of hydromechanical bifurcation Many components of the model can affect the capability of capturing unstable processes, such as the degree of nonassociativity, hydromechanical coupling (here introduced via the parameter r sw ), and the amount of inelastic wetting compaction (governed primarily by the plastic compressibility B p ). As previously pointed out, the role of coupling parameters in the prediction of these processes is unexplored. In order to focus on a limited set of material properties, and clarify their effect, it is useful to motivate the analyses through an analytical inspection of the constitutive equations. Consider for this purpose the stress strain response for water-undrained isotropic loading (an example of a stress path that involves changes in both stresses and suction). If net stresses are controlled, and the loading path is able to engage the plastic resources of the material, then the consistency condition requires _p s ¼ _p net þ _ss r þ s _ S r ¼ _p net þ C h _ Sr (16) where C h ¼ s þ (@ f R =@S r )S r is a coefficient that depends on suction, degree of saturation and retention curve (here considered to be given by an expression of the type s ¼ f R (S r )). By using equation (1), and assuming that plastic hardening involves only the volumetric strains (i.e. î s ¼, as in classical critical-state plasticity) it follows that _p net ¼ p s B p _å p v r swp s _ Sr C h _ Sr (17)

7 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 7 During water-undrained loading paths, the changes in the of shear strain localisation (equation (7)) converging asymptotically to zero (shear failure). Once the perturbation mode degree of saturation in equation (17) depend on the total volumetric strains (i.e. n S _ r ¼ S r _å v ). If plastic strains constitute the main contribution to the kinematic response (i.e. terised by a peak in shear stress (initiation of the instability). is changed to undrained shearing, the stress path is charac- _å v _å p v ) it can be stated that This is confirmed by the stability index for undrained simple _p net p shear (equation (1)), the evolution of which is plotted in s ð1 B h Þ_å p v (18) Fig. 7. p Even though typical model predictions of drained shearing exhibit a stable mechanical response, alternative forms of where instability can still be identified before critical state (i.e. before the stress state at which volumetric strains are no h ¼ B p S r n r sw þ C h p s (19) longer possible). Fig. 8 details the evolution of the stability indices I AS and I BS during drained shearing. The figure This simple analytical result shows that the presence of hydromechanical coupling alters the mechanisms through which the yield surface evolves. Changes in the degree of saturation, in fact, are no longer derived from imposed changes in suction, but are rather a consequence of the entire mechanical response. As a result, the net stress rate associated with plastic consistency (i.e. _p net ) is affected by the pseudo-softening term h : Such hydraulic-induced softening depends on retention properties (C h ), hydraulic hardening (r sw ) and plastic compressibility (B p ). In particular, the parameter B p exacerbates the potential for instability by amplifying the effect of the other hydromechanical terms. Since B p is notionally associated with the expected amount of plastic strains originated by wetting (Fig. ), these considerations suggest that soils particularly prone to wetting collapse tend to be characterised by a larger potential for hydromechanical instability. The set of model parameters adopted for the numerical simulations is given in Table 1. The parameters have been selected with the goal of reproducing the response of a loose, unsaturated sand. The value and the range of variation of the behavioural properties having a major effect on the stability indices (e.g. non-associativity, water retention parameters, hydromechanical coupling terms, etc.) have been assessed on the basis of the available data for a volcanic silty sand (Bilotta et al., ; Buscarnera, 1). Given the simplicity of 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 potential for wetting collapse on the slope stability scenarios, the effect of plastic compressibility has been explored in greater detail. MODEL PREDICTION OF HYDROMECHANICAL BIFURCATION MECHANISMS Simulation of material instabilities in unsaturated soils In this section, simple shear simulations are used to illustrate the capabilities of the theory. A convenient starting point is the comparison between drained and undrained shearing in saturated slopes. Fig. 7 shows two numerical simulations characterised by the same initial conditions (slope angle Æ ¼ 18, ó 9 î ¼ 3 kpa). By using the input parameters collected in Table 1, drained shearing produces a ductile (strain-hardening) response, with the stability index Drained shearing Undrained shearing Shear failure locus Peak in shear stress (initiation of instability) Shear failure 1 1 Normal effective stress, σ ξ 4 6 Normalised stability index α 1 In situ stress (after deposition) 3 I * AS (index for drained shearing) (index for undrained shearing) I * BS Shear failure (* I AS ) Undrained instability (* ) I BS Fig. 7. Analysis of response of saturated slope via simple shear tests simulations: stress paths for drained shearing and undrained shearing; evolution of stability indices for drained shearing and undrained shearing Table 1. List of parameters adopted for the numerical simulations Elastic parameters Yield surface parameters Plastic potential parameters Hardening parameters SWCC parameters Æ ¼. a f ¼. 63 a g ¼. 63 B p ¼ 1/r s ¼. 38 a R ¼. 43 k ¼. m f ¼ 1. 4 m g ¼ 1. 4 î s ¼. n R ¼ 1. 3 G ¼ 4 kpa M cf ¼. 9 M cg ¼ r sw ¼ m R ¼. p r ¼ 1 kpa M ef ¼. 7 M eg ¼ 1. 7

8 8 BUSCARNERA AND DI PRISCO 1 1 Region of latent instability I * AS (index for drained shearing) (index for undrained shearing) I * BS Shear failure ( I * AS ) Onset of latent instability ( * ) I BS Normalised stability index Fig. 8. Evolution of stability indices I AS and I BS during drained shearing: concept of latent instability shows that the condition I BS ¼ anticipates shear failure, defining a region in which a passage from drained to undrained conditions is critical. Since this passage is only potential, the onset of instabilities depends on the type of shear perturbation. As a result, the region of the stress space where I BS < corresponds to a domain of latent instability. Within such a region, model predictions suggest that flow instabilities are possible, and their potential occurrence is not overlooked if a convenient stability index is monitored. The natural extension of these ideas to partially saturated conditions relies on the comparison between constant-suction and water-undrained shearing. This section details only the characteristics of model predictions obtained within the unsaturated regime; the next section will expound the unstable transition from unsaturated to saturated conditions. In order to show the role of material properties, water-undrained shearing has been simulated by using two different values of plastic compressibility (Fig. 9). While the constant-suction scenario shares many similarities with the drained test previously simulated, the water-undrained scenario may resemble either drained or undrained shearing, depending on the initial 4 Constant-suction shearing (CS 1) Low compressibility ( B p 3) Water-undrained shearing (WU 1) Low compressibility ( B p 3) α 1 I * AU Stability index I * AU (test CS 1 ) Stability index I * BU (test WU 1 ) 3 1 Initial suction ( Sr ; s 7 kpa) Shear failure locus I * BU I * AU s rf I * AU (shear failure at constant suction) I * BU (instability at constant water content) In situ conditions (after deposition) 4 Constant-suction shearing (CS ) Low compressibility ( B p 9) Water-undrained shearing (WU ) Low compressibility ( B p 9) α 1 I * AU Stability index I * AU (test CS ) Stability index I * BU (test WU ) 3 1 Initial suction ( Sr ; s 7 kpa) I * BU s rf 78 I * AU (shear failure at constant suction) I * BU (instability at constant water content) In situ conditions (after deposition) 1 3 Normal skeleton stress, σ ξ (c) Normalised stability index (d) 8 1 Fig. 9. Analysis of response of unsaturated slope via simple shear test simulations. Low/moderate compressibility: stress paths and evolution of stability indices for constant-suction shearing and water-undrained shearing. Effect of high compressibility: (c) stress paths for constant-suction shearing and water-undrained shearing; (d) evolution of stability indices. S rf indicates degree of saturation at moment of failure

9 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 9 state and material properties. If the material is characterised by low or moderate compressibility, the evolution of the instability index I BU for water-undrained shearing is rather similar to that of constant-suction shearing, I AU : In this case failure is achieved on the same shear failure locus (Figs 9 and 9), and the degree of saturation undergoes minor changes. By contrast, high compressibility favours a response that is similar to the saturated/undrained scenario, with instability initiating at lower shear stresses (Figs 9(c) and 9(d)). In addition, in this case the degree of saturation undergoes more significant variations. The results disclose a remarkable increase in complexity compared to saturated conditions. The dependence of material stability on the saturation index makes it impossible to establish a direct correspondence between stress state and stability conditions. The latter depend on the incremental loading path and the evolving state of the material, with latent instability that tends to be predicted only for the loading paths that induce a non-negligible increase in the degree of saturation. Under this viewpoint, the volumechange properties of the soil can be crucial. In fact, when the suction and the degree of saturation do not change remarkably, suction-constant and water-undrained failure modes tend to coincide (i.e. I AU and I BU vanish simultaneously). By contrast, when soil compressibility is significant, S r changes, and the water-undrained failure mode can anticipate suction-constant localisation (i.e. I BU vanishes when I AU is still positive). In this case the failure scenario shares many similarities with the initiation of static liquefaction in saturated soils: it occurs when the soil has residual tendency to contract (i.e. when critical state has not yet been reached), and implies a peak in shear stresses. Stability indices can also support the analysis of unsaturated slopes when they are subjected to a more intuitive form of perturbation: the saturation promoted by rain infiltration. In this case, saturation tests with constant shear stress can be used to investigate stability conditions. In this work, the latter scenario is simulated by applying a removal of suction ( s, ) at constant net stress conditions. Fig. 1 shows three simulations in which the saturation stage is imposed at different stress conditions, corresponding to three different slope angles. The possibility of a localised shear failure is checked first (Fig. 1). At Æ ¼ 8 the value of I AU is always positive, given that the simulation does not exhibit any shear failure. Failure is reached at Æ ¼ 338 (i.e. very close to the angle of natural repose of the saturated soil), with the saturation index approaching zero at S r ¼ 1. Finally, the simulation performed for a steeper slope angle, Æ ¼ 48, exhibits failure in the unsaturated regime, with I AU vanishing when S r, 1. Even though the previous simulations are based on suction-controlled wetting paths (i.e. index I AU governs the initiation of failure), I BU can still be monitored through a latent instability analysis. Fig. 11 refers to the simulation of a suction-controlled saturation stage at Æ ¼ 38. Even if shear failure is not attained during the process of suction removal, instability processes are still possible, depending on soil properties. This is shown in Fig. 11, where I BU is monitored. The same simulation is in fact repeated by using increasing values of soil compressibility (i.e. by increasing the tendency to produce volume compaction upon suction removal). The simulations show that highly collapsible soils can suffer unexpected instability modes. The index I BU vanishes only for high values of soil compressibility, marking the attainment of an unstable state. It is worth noting that such predicted instabilities are only potential, being contingent to specific control conditions. In other words, the model would predict an actual collapse of the slope only when the system is perturbed in a certain manner (e.g., in Normalised stability Index, I * AU Initial suction ( S ; s r 7 kpa) I * AU 1 Shear failure locus Onset of shear failure 3 Normal skeleton stress, σ ξ α 4 Initial degree of saturation in situ α Degree of saturation, α this case, by imposing water-undrained conditions or injecting water volume). Unstable transition from unsaturated to saturated conditions This section discusses some particular types of model prediction that are associated with the unstable transition from unsaturated to fully saturated conditions. In other words, it illustrates a mechanical scenario that takes place under partially saturated conditions and provides a justification for catastrophic liquefaction failures. Fig. 1 illustrates a series of numerical simulations of water-undrained shearing starting from initially unsaturated conditions. Increasing values of compressibility are associated with larger changes in suction upon loading. This effect promotes a general shift of the stress paths to the left, thus anticipating the prediction of instabilities. The sequence of failure mechanisms can be clarified by inspecting the evolution of the stability indices. For relatively low values of plastic compressibility there is no prediction of local peaks in shear stress (simulations WU 3 and WU 4 in Fig. 1, and WU 7 and WU 8 in Fig. 1), being shear failure achieved on the critical-state line (I BU ¼ I AU ¼ ). Local peaks in shear stresses can instead 9 S r α 4 α 33 1 α 4 Fig. 1. Simulation of the response of unsaturated slope by simple shear tests: stress paths for three saturation tests at constant shear stress; evolution of stability index for failure under constant suction (I AU ) 1 1

10 1 BUSCARNERA AND DI PRISCO Normalised stability Index, I * BU Initial suction ( S ; s r 7 kpa) 1 6 I * BU Shear failure locus : Latent instability 3 4 Normal skeleton stress, σ ξ Initial degree of saturation 7 8 Degree of saturation, be predicted for larger values of compressibility (simulations WU and WU 6 in Fig. 1, and WU 9 and WU 1 in Fig. 1), and are associated with the fulfilment of I BU ¼ prior to I AU ¼ (closed square symbols in Fig. 1). It is interesting to note that the value of degree of saturation at failure (S rf in Fig. 1) depends remarkably on the plastic compressibility and the associated water-undrained stress path, and should therefore be considered as a path-dependent characteristic. If total stresses are assumed to be controlled, a peak in the shear stress is associated with the prediction of an uncontrolled saturation of the pores. It is therefore particularly interesting to monitor the evolution of the index I BS during these simulations. In fact, since the post-peak volumetric response keeps being contractive, continued shearing causes a further increase in S r and full saturation (open square symbols in Fig. 1). If saturation conditions are established, I BS becomes the relevant index for undrained shearing, and affects the predictions upon continued undrained shearing. Thus very different types of post-peak response can be envisaged. In some cases the index I BS is negative at S r ¼ 1 (simulations WU,WU 6 and WU 9 ), thus having a continued decrease in shear stresses. Other simulations are instead characterised by a recovery of shearing resistance upon continued shearing and two successive peaks in shear stress (simulation WU 1 ). The latter circumstance 9 3 S r α B p 3 B p B p 7 B p 9 Latent instability: I * BU Fig. 11. Possibility of latent instabilities during saturation; evolution of stability index for wetting collapse (I BU ) Symbols Condition I * AU S rf 87 Initial suction ( Sr 7; s 7 kpa) Initial suction ( S 7; s r 7 kpa) I * BU I * BS S r 1 WU 6 B p 1 WU 1 B p 1 S rf 87 WU B p 8 S rf 93 I * BS at saturation WU 4 B p 6 S rf Normal skeleton stress, σ ξ WU 9 B p 8 S rf 96 WU 8 B p 6 S rf 93 I * BS 1 1 Normal skeleton stress, σ ξ can be predicted if the saturation point is attained at relatively small values of mobilised friction angle (i.e. at shear stresses that do not yet correspond to spontaneous liquefaction). As illustrated in Fig. 13, this possibility implies a recovery in resistance ( ô), with a predicted strength capability that can even overcome the shear stresses responsible for the activation of the first bifurcation mechanism. For this reason, such simulated instability modes can be interpreted as predictions of metastable states that is, situations at which the development of an unstable mode of deformation is interrupted by the transition from unsaturated to fully saturated conditions. Metastability can be also predicted during saturation paths. Again, parallel monitoring of both the liquefactionrelated stability index and the wetting-induced bifurcation condition is important for identifying these mechanical conditions. Fig. 14 shows two simulations of saturation paths at constant shear stress. While larger values of the slope angle imply that the stress threshold for incipient liquefaction is crossed during wetting (I BS ¼ in Figs 14 and 14(c)), gentle inclinations are not associated with a state of incipi- α WU 3 B p 4 S rf 8 14 α WU 7 B p 4 S rf 8 I * BS Fig. 1. Simulation of water-undrained simple shear mechanisms: effect of soil compressibility on unstable transition from unsaturated to saturated conditions

11 Second bifurcation: I * BS SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 11 Sr 7 s 7 kpa B 1 p Saturation of the pores: I * BS 1 1 Normal skeleton stress, σ ξ :kpa First bifurcation: I * BU Fig. 13. Numerical prediction of metastable conditions: first peak (bifurcation under unsaturated conditions, I BU ) causes full saturation, and is followed by second bifurcation point (second peak at I BS ) at larger stresses ( ô indicates the predicted recovery in shearing resistance) ent liquefaction (I BS. in Figs 14 and 14(d)). This difference is fundamental in grasping the effects of wetting collapses initiated by suction removal. While an unstable transition from unsaturated to saturated conditions is the trigger of a subsequent liquefaction in the first case (I BS, when S r ¼ 1 in Fig. 14(e)), this is not the case in gentle slopes, given that possible wetting collapses are transient metastable conditions, not necessarily associated with a subsequent liquefaction of the layer (i.e. shearing resistance can be recovered after the first bifurcation mechanism; Figs 14 and 14(f)). STABILITY CHARTS OF HYDRAULIC PERTURBATIONS: EFFECT OF SUCTION REMOVAL The stability of unsaturated deposits during rainfall events can be studied by simulating the response of the slope to wetting paths. Although the complete quantification of suction perturbations over time (as well as the associated rainfall thresholds for slide triggering) would require data from transient rain infiltration analyses, it is possible to simplify the description of the hydrologic effects by representing their perturbations as suction removal processes (i.e. s, ). In this way the changes in suction are closely related to the disturbance effectively altering the current state of the slope during a rainfall event, and are likely to be associated with the onset of material failure and the consequent activation of a slide. The material point simulations illustrated in the previous section can be used to condense the effect of material parameters and slope characteristics (e.g. deposit thickness, slope inclination). In other words, the perturbations able to induce an instability can be identified through the stability indices obtained from material point analyses. Such simulations are therefore used to define instability scenarios for given sets of initial saturation conditions, slope inclinations and types of disturbance. The outcome of these analyses is eventually collected in graphical charts of triggering perturbations, hereafter referred to as stability charts. It is worth noting that, although the focus of this paper is on the mechanical implications of suction removal processes, such Δτ 3 a strategy can in principle be combined with more sophisticated retention models and with advanced hydrologic descriptions of the infiltration process, therefore studying the effect of prior infiltration/evaporation events. The stability charts discussed in this section will be based on suction-controlled wetting tests, and will be presented in terms of changes in suction. In order to compare different initial conditions and have a more convenient graphical representation, triggering perturbations are normalised for the initial suction, s : In this way, full saturation is achieved when the normalised perturbation s ¼ j sj=s ¼ 1: The possibility of shear failure induced by saturation paths is investigated first. Fig. 1 shows the stability charts obtained by studying the evolution of I AU for different initial saturation conditions. All charts converge to the same point, coinciding with the angle of natural repose (Æ NR ) of the saturated material. For that inclination, in fact, failure is obtained only when suction is completely removed. The stability charts are not defined in the range of slope angles lower than the angle of natural repose, for which the contribution of suction is not necessary to ensure stability. By contrast, the value of suction is critical to assess stability conditions when Æ. Æ NR, as higher suctions enable the material to sustain steeper inclinations. Fig. 1 illustrates the practical use of these stability charts. Once the in situ state is defined, the associated chart provides the magnitude of triggering perturbation for any slope inclination. It is then possible to evaluate this critical change in suction for a deposit of a given inclination. Shear failure is the most intuitive type of instability expected during saturation. As previously shown, however, other forms of instability are possible upon suction removal. This possibility is explored in Fig. 16, where stability charts of the index I BU are reported. Any point of the chart is associated with condition I BU ¼ : Thus the charts have been obtained by controlling the evolution of the stability index associated with water-content control during the process of suction removal. As in the previous examples, the effect of soil compressibility is investigated. The effect of this parameter on the stability charts is remarkable. Whereas for relatively stiff soils there is practically no difference with the stability chart of shear failure, higher values of compressibility alter significantly the predicted stability scenario. The charts derived from I BU indeed tend to shift below those associated with I AU : In other words, the numerical results reflect the fact that the state of the slope has entered a domain of latent instability, and suggest that a multiplicity of failure modes can be predicted for highly collapsible materials (i.e. volumetric instabilities can arise before the attainment of shear failure). These considerations can be linked to the notion of metastability discussed in the previous section. Depending on the specific material parameters, the prediction of mathematical bifurcation can indeed assume different connotations, which in turn reflect distinct forms of wetting instability. As was indicated in the previous section, information on metastable states must be derived from the combined analysis of indices I BU and I BS : For example, for high values of compressibility the stability chart associated with I BU ¼ shifts below the chart of I BS ¼ (Fig. 17). This circumstance reflects the possibility of a metastable condition (as illustrated in Fig. 14), and does not suggest a spontaneous sequence of unstable mechanisms. On the contrary, when the slope angles prone to latent instability lie within the range of potential liquefaction (i.e. I BS < ), the model predicts the possibility of two successive instability mechanisms leading to liquefaction (Fig. 17). In other words, if I BU vanishes when the stability boundary for flow failure has already been crossed, wetting instability should be regarded as a precursor for

12 1 BUSCARNERA AND DI PRISCO Initial suction ( Sr 7; s 7 kpa) Symbols Condition Shear failure I * AU I * BU I * BS S r 1 Attainment of full saturation P α 33 Change in control 1 1 Initial suction ( S 7; s 7 kpa) r Attainment of full saturation α P 1 Change in control Normalised stability indices, I * Normal skeleton stress, σ ξ I * BU at P I * AU I * BS at P P (change in control) I * BU I * BS α Normal skeleton stress, σ ξ 1 33 α 1 1 I * BS I * BU I * AU I * BS at P I * BU at P P (change in control) 3 Normalised stability indices, I * I * BS Degree of saturation, S r (c) P (change in control) α Degree of saturation, S r (d) 1 33 α 1 I * Water-undrained BS deformation stage at S r 1 I * BU Degree of saturation, S r (e) I * BU I * BS Water-undrained deformation stage P (change in control) Degree of saturation, S r (f) I * BS at 1 S r Fig. 14. Numerical simulation of unstable transition from unsaturated to saturated conditions and metastable response (B p.1); (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 liquefaction, given that the initiation of unstable saturation is followed by a continuous reduction in shearing resistance. In the light of these analyses, the achievement of fully saturated conditions can be the consequence of an unstable process rather than a prior cause of collapse, and the liquefaction event can be understood as the ultimate stage of a chain process activated by a prior suction removal. The theoretical interpretation of the numerical simulations allows two opposite scenarios to be distinguished. The first one refers to soils whose volumetric response is rather insensitive to wetting paths. In this case, there is no practical difference between I AU and I BU (both providing the same prediction of failure; Fig. 18), and shear strain localisation is the only mechanism that can be initiated by

13 1 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 13 1 Normalised suction perturbation, Δs* Δs / s Normalised suction perturbation, Δs* Δs / s No localised shearing failure upon saturation S r 8 s 4 S r 9 s 1 7 S r 8 In situ state: α 3 γz 3 kpa S 8 s r 4 kpa Slope angle, α: degrees S r 7 s 7 S r 6 s 14 suction removal. This scenario is possible for slope angles larger than the angle of natural repose of the saturated material (interval in Fig. 18). The second scenario refers to soils exhibiting significant volume collapse upon wetting. In this case, hydromechanical coupling can be critical, and yields the existence of additional instability modes. The condition I BU ¼ originates a distinct stability chart, located below the stability domain associated with I AU ¼ (Fig. 18). The extent of the range of slope angles 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 critical slope angles is found (interval ), for which wetting collapse can dominate the failure response of the slope. The main outcome of this scenario is an extension of the range of unstable slope angles and, most notably, a change in the features of the expected instability mechanism. The predicted wetting-instability modes can be further differentiated on the basis of the value of the index I BS for saturated-flow failure (Fig. 19). In all cases, the onset of a wetting instability under total stress control (I BU ¼ ) has the effect of inducing full saturation (i.e. s ¼ 1, as illustrated by the vertical arrows in Fig. 19). After this process, different scenarios can be devised. In particular, when the stability charts obtained from the index I BS Increase in S r γz 1 kpa α NR Slope angle, α: degrees No localised shearing failure upon saturation I * AU : shear failure upon suction removal Fig. 1. Stability charts of hydraulic perturbations for shear failure (charts obtained by checking occurrence of I AU : role of initial saturation conditions; example of use of chart for given in situ conditions α NR Normalised suction perturbation, Δs* Δs / s Normalised suction perturbation, Δs* Δs / s S s r 8 4 kpa γz 3 kpa B p 3 B p B p 7 B p 8 1 B p 3 B p B p 7 B p 8 S s r 8 4 kpa γz 3 kpa 1 Increase in soil compressibility 1 Slope angle, α: degrees I* AU I* BU : shear failure upon suction removal I * : possible latent instability (volume collapse can take place upon suction removal) BU In situ state: α 3 S 8 s r 4 kpa 1 Slope angle, α: degrees Fig. 16. Stability charts of hydraulic perturbations for latent instability during suction-controlled saturation (charts obtained by checking occurrence of I BU ). Role of soil compressibility in the possibility of entering a region of latent instability indicate that the boundary for a possible isochoric flow are crossed before the fulfilment of condition I BU ¼, the wetting instability takes place within a domain that is already prone to static liquefaction (i.e. I BS < ). In this case, wetting instability can be regarded as the hydromechanical trigger of a flow instability. By contrast, when the unstable saturation of the pores coincides with I BS >, it reflects the existence of a metastability domain in which condition I BU ¼ is no longer a precursor of catastrophic liquefaction (interval a in Fig. 19). CONCLUSIONS This paper has detailed the study of flow slides triggered by rainfall. The aim has been to provide a modelling framework for explaining failure in unsaturated slopes and performing triggering analyses. For this purpose, the scheme of an unsaturated infinite slope has been combined with the basic principles of unsaturated soil elasto-plasticity and the mathematical concept of controllability. In this way, the response of the deposit has been reproduced by means of simple shear simulations, and analytical indices have been used to study the stability of the slope

14 14 BUSCARNERA AND DI PRISCO Normalised suction perturbation, Δs* Δs / s Normalised suction perturbation, Δs* Δs / s Δs for I* S s r 8 4 kpa γz 3 kpa B 7 p BS Δs for I* Range of slope angles for wetting instability: I* and I* BU 3 Slope angle, α: degrees Δs for I* AU Δs for I* Δs for I* BS BU 3 Slope angle, α: degrees BS Range of slope angles susceptible to metastability: I* and I* BU BU BS Δs for I* 3 3 S s r 8 4 kpa γz 3 kpa B 8 p AU Fig. 17. Stability charts for wetting instability (I BU ) and liquefaction (I BS ): stability threshold for incipient liquefaction already crossed at I BU (unstable transition from wetting instability to flow failure); at low angles I BU anticipates conditions of incipient liquefaction (I BS > and possible metastability is predicted at S r 1) 4 4 Δs / s Δs / s Latent instability I * BU 1 The analyses show that wetting paths can trigger a multiplicity of unstable phenomena, and that some of these instabilities can anticipate shear failure. Three types of mechanism have been studied: localised shear failure, static liquefaction and wetting collapse. In particular, it has been shown that the unstable mode associated with wettinginduced collapse shares several features with static liquefaction. The major difference, however, is that wetting-collapse phenomena are predicted to occur when the material is not yet saturated, and can therefore be activated by the process of suction removal. According to this interpretation, saturated conditions may not be necessary to trigger a flow slide, being liquefaction potentially originated from a chain process consisting of volume collapse, uncontrolled saturation and, eventually, catastrophic undrained failure. These analyses point out that the combined use of several stability indices is critical for distinguishing different failure scenarios. They also emphasise the importance of using a unified strategy of analyses, in which shear failure, saturationinduced liquefaction and metastability are all naturally included as particular cases. In order to highlight the engineering significance of these notions, stability charts representing the triggering perturbations as a function of the slope inclination have been numerically evaluated. The analyses show that the possibility of undergoing volumetric instabilities also depends on the parameters that introduce hydromechanical coupling. This conclusion suggests that rainfall-induced flow slides are exceptional phenomena that can take place only in very particular deposits, susceptible to both undergoing liquefaction and experiencing volume compaction upon saturation. Most notably, these results suggest that the risk of rainfall-induced flow slides may depend on material properties that are not directly associated with the shearing resistance. In order to clarify these concepts, parametric analyses have been performed. Such analyses allowed assessment of the relation between the parameters governing wetting-induced compaction and the range of slope inclinations affected by the instability mechanisms. It has been found that the instability of slopes made of materials that are insensitive to wetting paths is dominated by shear failure, liquefaction being possible only under fully saturated conditions. In contrast, soils that are highly collapsible upon wetting are associated with a much broader spectrum of unstable responses, which also includes the initiation of liquefaction upon either suction removal or saturation-induced metastability. Localised shear failure I * AU I* BU Shear failure I * AU Fig. 18. Schematic representation of stability charts. Waterinsensitive soils: shear strain localisation is only failure mode (I AU and I BU provide the same results). Relevant tendency to collapse on wetting (hydromechanical coupling). Latent instability is distinguished from shear strain localisation (I AU and I BU do not provide the same results) 3 α α

15 Δs / s Δs / s 1 1 SOIL STABILITY AND FLOW SLIDES IN UNSATURATED SHALLOW SLOPES 1 8 _ó 9 Saturation instability ( S 38 9 r 1) î D 11 D 1 D 13 D 14 D 1 D 16 D 17 _å activates liquefaction î _ó ç D 1 D D 3 D 4 D D 6 D 7 _å ç >< _ó D 31 D 3 D 33 D 34 D 3 D 36 D 37 >= _å >< >= Static _ô îç ¼ D 41 D 4 D 43 D 44 D 4 D 46 D 47 _ª liquefaction îç Latent instability I * BS I * _ô ç D 1 D D 3 D 4 D D 6 D 7 BU _ª ç 6 _ô î 4 D 61 D 6 D 63 D 64 D 6 D 66 D 67 7 _ª î >: >; >: _s D 71 D 7 D 73 D 74 D 7 D 76 D 77 S _ >; r () Although many other factors that can affect hydromechanical stability (e.g. coupled retention properties, hydraulic history) have not been specifically addressed in this paper, the theoretical methodology can be enhanced for quantifying their effect. At variance with the usual stability analyses for unsaturated slopes, the proposed approach is based on stability indices that reflect the role of suction removal processes and enable multiple failure mechanisms to be simultaneously accounted for. As a result, in a future perspective this theory can be combined with detailed experimental characterisation of unsaturated deposits and hydromechanical models calibrated for site-specific features. In the authors opinion, the achievement of such a coordinated effort between geomechanical modelling and geotechnical site characterisation can constitute a powerful tool for locating areas prone to originate slope failures and estimating the risk of activation of flow slides. APPENDIX This section briefly illustrates the analytical derivation of the stability indices used in this paper. More details are available in Buscarnera & di Prisco (11b). Equation () reports the complete incremental hydromechanical response of the material points within a slope. Saturation instability ( S r 1) Metastability Zone of possible metastable evolution a I * BS Liquefaction Latent instability * I BU Fig. 19. Schematic representation of instability charts. Stability chart for I BS is entirely located below condition for I BU ; static liquefaction can be the ultimate consequence of a wetting instability. Stability chart for I BU can be above the condition for I BS (interval a); evolution of wetting instability can be characterised by condition of metastability (shaded area indicates zone of possible metastable evolution) b α α The kinematic constraints deriving from the assumption of an infinite slope can be used to simplify the above relations. Planestrain conditions imply that _å ¼ _ª ç ¼ _ª î ¼, while the symmetry along the ç axis implies _å ç ¼ : These kinematic constraints are representative of a simple shear strain mode, and lead to equation (3). The stability index for mode A can be obtained by considering an incremental loading path characterised by controlled changes in stresses and suction (e.g. constant-suction shearing, constant-stress suction removal). In this case the control variables coincide with those collected in the left-hand side of equation (3). The theory of test controllability therefore identifies the inception of a bifurcation mode when the constitutive matrix in equation (3) is singular: that is, when D 77 ðd 11 D 44 D 14 D 41 Þ ¼ (1) in which conditions D 71 ¼ D 74 ¼ have been used. Mode B can be derived by modifying equation (3) to reproduce water-undrained loading. If the drainage of the water phase is prevented, the evolution of both degree of saturation and suction depends on the overall mechanical response of the material. Under constant water content, the constraint relating volumetric strains and saturation index is given by n S _ r ¼ S r _å î () By expressing _ó î as a function of the increment in total normal stress and suction (i.e. _ó î ¼ _ó î þ S r _s), the constitutive equations can be reformulated as _ó î _ô îç where ¼ D 11 D 14 D 41 D 44 D 11 ¼ D 11 S r n _å î _ª îç D 17 S r n D 77 (3) (4) D 41 ¼ D 41 S r n D 47 () in which the hydraulic variables are eliminated by using the constraint in equation (). Equations (3) are complemented by n_s ¼ (S r =n)d 77 _å î, which is needed for tracking the changes in suction during shearing. It is interesting to note that the stability index associated with failure of the slope under constant water content coincides with the determinant of the control matrix in equation (9). NOTATION a f shape parameter of the yield surface a g shape paremeter of the plastic potential a R shape parameter of the retention curve B p plastic compressibility parameter C h hydraulic parameter related to retention properties D hydromechanical constitutive matrix D ij principal minors of the constitutive stiffness matrix _E hydromechanical strain vector e void ratio e w water ratio f R soil water retention curve G elastic shear modulus I AS stability index for shear failure (saturated soil) I AU stability index for shear failure (unsaturated soil) stability index for liquefaction (saturated soil) I BS