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Landslides involving sensitive clays are recurrent events in the world’s northern regions and are especially notorious in eastern Canada. The two critical factors that separate sensitive clay landslides from traditional slope stability analysis are the highly brittle behavior in undrained conditions (strain-softening) characteristic of progressive or retrogres- sive failures and the large deformations associated with them. Conventional limit equilibrium analysis has numerous shortcomings in incorporating these characteristics when assessing landslides in sensitive clays. This paper presents an extensive literature review of the failure mechanics characteristics of landslides in sensitive clays and the existing constitutive models and numerical tools to analyze such slopes’ stability and post-failure behavior. The advantages and shortcomings of the different techniques to incorporate strain-softening and large deformation in the numerical modeling of sensitive clay landslides are assessed. The literature review depicts that elastoviscoplastic soil models with non-linear strain-softening laws and rate effects represent the material behavior of sensitive clays. Though several numerical models have been proposed to analyze post-failure runouts, the amount of work performed in line with sensitive clay landslides is very scarce. That creates an urgent need to apply and further develop advanced numerical tools for better understanding and predicting these catastrophic events. Keywords Progressive landslide, Sensitive clay, Numerical modeling, Strain-softening, Constitutive soil model, Large deformation among which 134 fatalities are recorded solely in the Introduction Québec region due to the glaciomarine-sensitive clay fail- Landslides in sensitive clays are recurrent events in the ures in the St. Lawrence Lowlands (Blais-Stevens 2019). northern countries of the world, especially in Canada Because of the nature of sensitive clay, the runout and and Norway. The impact of landslides is catastrophic to affected area of these landslides are generally very large. both the population and the economy. Natural resources In the sensitive clays of Norway, the Gjerdrum landslide Canada (NRCan) has reported that in Canada, the annual (2020) spanned a flow-off area of 210,000 m and addi- damages by landslides are worth $200 to $400 million tionally affected 90,000 m by debris flow. A total of (NRCan 2019). A total of 778 people have died in land- 1000 people were evacuated, ten people died, 31 houses slide events all over the country from 1771 to 2018, were destroyed, and the mitigation works were worth $20 million without the rebuilding cost of infrastructure *Correspondence: or environmental damages (Liu et al. 2021). Therefore, Zinan Ara Urmi understanding the triggering factors, failure mechanisms, email@example.com and post-failure consequences of these landslides is vital Université du Québec à Chicoutimi, 555 Bd de L’Université, Chicoutimi, Saguenay, QC G7H 2B1, Canada for risk assessment and improving the resiliency of the Virginia Tech, Blacksburg, VA 24061, USA affected communities. © The Author(s) 2023. 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Geoenvironmental Disasters (2023) 10:14 Page 2 of 28 Understanding how material sensitivity controls soil The subsequent sliding is referred to as "progressive" if behavior is of utmost importance in analyzing landslides the failure progresses forward and "retrogressive" if the in sensitive clays. The unique nature of sensitive clays is failure propagates rearward (Varnes 1978). that under shear loading, after reaching the peak shear The current state of knowledge related to the detailed strength, there is a dramatic reduction in shear strength analysis of landslides in sensitive clays depicts that with increasing strain (Pusch 1966); this phenomenon computational modeling is more adaptable than other is generally referred to as "strain-softening." In this con- approaches, such as theoretical analysis, empirical anal- text, "sensitivity" is defined as the ratio of the peak shear ysis, and experimental investigations. Conventional strength to the reduced shear strength that expresses the limit equilibrium analysis has numerous shortcom- loss of strength when the soil experiences large deforma- ings in incorporating complex geometry, non-linear soil tion (Crawford 1968; Skempton and Northey 1952). The behavior, and real-life field condition. Moreover, it can - development of sensitivity of the clays is attributed to not predict post-failure runout. Alternatively, numeri- the depositional features of sensitive clays as well as the cal modeling is a handy tool for slope stability analysis ongoing weathering effect on embankment soils. Sensi - because it requires fewer assumptions, especially regard- tive clays are believed to be deposited in marine environ- ing the failure mechanism, and can account for complex ment depressions left by the Laurentian ice sheet around constitutive behaviors. Even though numerical mod- 14,000 to 6000 years ago from the present time (Quig- eling for landslides has come a long way in the last three ley 1980; Lefebvre 1996, 2017). Due to the exposure to decades, a few works have focused on the challenging seawater with high salt concentration, the clays formed features of sensitive clays (e.g., strain-softening, the tran- a flocculated structure with high undisturbed shear sition from solid to liquid form, and post-failure large strength. With the deglaciation of the ice sheets over deformation) (Dey et al. 2015; Tran and Sołowski 2019; time, the lands which were once depressed by the huge Wang et al. 2022). The reasons for this lack of information weight of the ice sheets rose above the sea level (iso-static are, for example, that the simulation of strain-softening rebound). Due to this uplift of the clay deposition above materials is challenging in continuum numerical frame- the seawater, the clays got exposed to fresh water. When works, strains tend to develop and localize along narrow the freshwater flows through the soil, the salt concen - shear bands, and several mesh-regularization techniques tration within the soil mass reduces due to the leaching need to be adapted to obtain mesh-independent results out of salt into the fresh water. As a result, even though (Rødvand et al. 2022; Singh et al. 2021; Thakur 2011). the clays retain their flocculated structure, they do not In addition, a well-established constitutive framework have the salt ions that were keeping the structure stable. is required to capture the transition from solid to liquid This structure is called meta-stable, which is highly sus - behavior. Finally, the modeling of post-failure behavior ceptible to disturbance and leads to very low remolded in history-dependent materials is complex, and current strength. Some marine clay deposits exhibit exception- state-of-practice numerical techniques (i.e., finite ele - ally high sensitivity after the significant reduction in ments and finite differences) suffer from mesh tangling shear strength with increasing strain; the soil completely when dealing with large deformations (Sulsky et al. 1994). loses its structural stability and transforms from a solid The objective of this paper is to provide an extensive to a liquid-like substance (Rosenqvist 1953, 1966). This review of (a) the behavior of sensitive clays under shear process is termed "remolding of sensitive clays," and loading, (b) the landslide mechanisms in sensitive clays, the shear strength at which the process of remolding and (c) the available numerical models (i.e., constitutive begins is termed the "remolded shear strength." These laws and numerical frameworks) used for the assess- soil deposits are also referred to as "quick clays." Sensi- ment of such landslides by examining the concerns and tivity values for quick clays are generally greater than 30 advancements of each technique. The paper is organ - with remolded shear strength less than 0.5 kPa (Lefebvre ized as follows. First, the post-peak stress–strain behav- 1996). Crawford (1968) reported that eastern Canadian ior of sensitive clays is presented. Then, typical sensitive quick clays have a sensitivity value ranging from 20 to clay landslides’ triggering and failure mechanisms (i.e., several hundred. In a sensitive clay landslide, the initially flows and spreads) are outlined. After that, the existing deformed soil deposits get remolded and flow away from numerical tools used for the evaluation of sensitive clay the source area, leaving the newly formed slope unsup- landslides are revised. In particular, the specific util - ported, which may initiate another instability. This pro - ity of constitutive models that address strain-softening cess can lead to a series of failures extending far beyond and numerical frameworks that cover large deformation the crest of the initial slope (Bjrrum 1955). Hence, for problems in the assessment of landslides are discussed. highly sensitive clays, the slope failure initiation is not the Finally, the literature review is compiled in a summary only concern, but the post-failure analysis is also critical. Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 3 of 28 table, and the conclusions and future research lines are Early studies on the stress–strain behavior of sensi- highlighted. tive clays explicated strain-softening phenomenon based on experimental results (Skempton 1964; Skempton Behavior of sensitive clays and Northey 1952). Skempton (1964) illustrated that if When subjected to monotonic shear loading in drained overconsolidated clays are strained, the shear strength conditions, sensitive clays have a robust collapsible would initially increase up to a certain point (peak shear nature for both normally consolidated (NC) or over con- strength). Then, the strength would start to decrease solidated (OC) state. Due to the depositional history of gradually with increasing strain to a residual value at a sensitive clays, during drained shear failure, they expe- large displacement. He stipulated that post-peak strain- rience a dispersion in their meta-stable structure and a softening occurs due to the reduction of the effective simultaneous decrease in porosity (Enyang et al. 2019). cohesion (c’) and friction angle (φ′) of such clays to a In contrast, the non-sensitive clays show dilative or con- residual value (c’ and φ′ ) (Fig. 1). Bjrrum (1961) sug- r r tractive nature in drained shearing based on being OC gested that in undrained shearing, increasing pore pres- or NC, respectively. This difference in volumetric behav - sure with increasing strain might cause a decrease in ior in drained conditions directly impacts the undrained shearing resistance due to the diminished effective stress. behavior of clays. Although no volumetric deformation Consistent with this idea, several researchers in the last is expected in undrained conditions, the material’s ten- two decades stipulated that the post-peak shear strength dency to contract or dilate governs the sign of the excess reduction in soft sensitive clays is governed by shear- pore pressure (positive or negative, respectively) gener- induced pore pressure rather than by a reduction of the ated during the loading process, impacting the undrained values of the strength parameters (φ′ and c′) (Bernander shear strength and the soil behavior. While non-sensitive 2000; Gylland et al. 2012; Thakur 2011, 2007; Thakur et al. OC clays generate negative pore pressure (suction) and 2006). Figure 2 represents the stress–strain relationship experience enhanced undrained shear strength (with and stress paths of CU triaxial testing, where the und- respect to the drained condition), non-sensitive NC clays rained effective stress path (ESP) follows a unique failure generate (positive) excess pore water pressures leading to line when subjected to undrained shearing (Thakur et al. reduced undrained shear strength. Mild strain-softening 2014). The resulting undrained strain-softening behavior behavior might be observed in either case, but it does not is related to the increasing shear-induced pore pressure greatly impact shearing resistance. On the contrary, in (P ), thereby reducing the effective stress. Thakur et al. sensitive clays (NC or OC), the strong tendency to col- (2014) supported that reductions in φ′ and c′ are possible lapse leads to a massive generation of excess pore pres- when sensitive clays are subjected to very large strains, sure, the shearing resistance is reduced to a negligible which was demonstrated in constant volume ring shear value, and the strain-softening behavior is exacerbated test results on low-sensitive Drammen plastic clay (Stark in comparison with the one experienced by non-sensi- and Eid 1994). tive clays (Lefebvre 1981). Both non-sensitive and sensi- Canadian sensitive clays can have a remolded und- tive OC clays exhibit similar strain-softening behavior in rained shear strength s less than 1.5 kPa measured by ur drained conditions. Swedish fall cone tests (Demers et al. 2014; Locat et al. Fig. 1 Drained behavior of an overconsolidated clay under shear loading (after Skempton 1964) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 4 of 28 Fig. 2 Illustration of the strain-softening mechanism. a Development of excess pore water pressure resulting in reduced shear strength; and b Eec ff tive stress path and total stress path following a unique failure line indicating no reduction in effective cohesion and friction angle (after Thakur et al. 2014) 2017, 2015). Tavenas et al. (1983) determined the strain of time-dependent strain, generally known as creep. energy required to remold sensitive clays up to a certain Creep is a phenomenon in which a soil mass undergoes a percentage (75–90%) for different locations in eastern slow and gradual deformation over time while subjected Canada and plotted the remolding index (I ) vs. strain to constant effective stress (Yin et al. 2011). It was noted energy curves. Later on, Quinn et al. (2011) converted that low strain rates caused increased compressibility those curves to stress–strain curves and pointed out that and reduced shear strength. Other studies on sensitive the shear strain required for 75%-90% of remolding is far Leda clay also support the fact that the undrained shear beyond 100% (Fig. 3a). Stark and Contreras (1996) also strength of the soil increased by about 6–12% for a ten suggested that a wholly remolded state may occur when times increase in strain rate (Graham et al. 1983; Leroueil the specimen is sheared to several hundred millimeters, et al. 1985). Lefebvre and Leboeuf (1987) found a linear equivalent to several hundred percent shear strain in soil variation of shear strength with the logarithm of strain specimens subjected to conventional laboratory tests. rate by performing several monotonic and cyclic triaxial But such large strains cannot be attained by standard lab- tests on three undisturbed sensitive clay samples from oratory shear testing. Stress–strain curves obtained from eastern Canada (Fig. 4). triaxial compression tests are limited to only 10–20% Field observations of sensitive clay landslides have indi- strain (Thakur et al. 2014). Generally, ring shear tests, cated that the creep behavior significantly influences the reversal shear box tests, or direct simple shear tests are field conditions for landslide initiation in glacimarine- used to study the large deformation behavior, but these sensitive clays (Okamoto et al. 2004). Creep development laboratory-scale shear tests could only achieve strains up leading to failure has three stages, a primary stage when to 30–45% (Durand 2016; Locat 2007; Locat et al. 2017, the strain rate decreases over time, a secondary stage 2015, 2013). This limitation warrants the need for strain- where the strain rate is constant, and a tertiary phase softening equations to predict the complete post-peak when the change in strain rate accelerates until global softening behavior up to the remolded stress (Urmi et al. failure occurs (Okamoto et al. 2004; Yin et al. 2011). 2022). Figure 3b illustrates DSS results in undrained con- Neglecting the impact of strain rate during the primary ditions for sensitive clays from five different locations in stage can promote the initiation of failure. However, dur- Canada; it can be observed that strain-softening in sensi- ing the tertiary phase, not considering strain rate can lead tive clays is highly non-linear. to overestimating the velocity of flowing debris, resulting The behavior of sensitive clays is significantly affected in more significant retrogression and runout. by the deformation rate. Vaid et al. (1979) performed In sensitive clays, the sliding surface may develop in a one-dimensional and isotropic consolidation triax- narrow shear band due to the strain-softening response ial tests on heavily consolidated sensitive clay. They before any significant movement occurs. The strength observed that the compressibility and the undrained degradation process is initiated when the gravitational shear strength considerably depend on the development shear stress surpasses the peak strength. As a result, Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 5 of 28 (a) Saint-Jude 22.7m Saint-Monique 13.45m Saint-Barnabé-Nord 23.31m (b) Casselman 22.3m Saint-Luc-de-Vincennes 20.2m Shear Strain (%) Fig. 3 a Analytically interpreted stress–strain curves based on the data of the remolding index vs. normalized strain energy curves by Tavenas et al. (1983) (after Quinn et al. 2011) b Experimental DSS test results showing strain-softening behavior of sensitive clays for different locations in Canada; the legends show the depth of the sample along with the location name ( after Durand 2016; Locat 2007; Locat et al. 2017, 2015, 2013) unbalanced stresses are transferred to the surrounding propagation of the shear band is known as the charac- areas, potentially overstressing the neighboring points. teristic length or the critical length of the shear band. This process eventually leads to the development of a Several analytical and numerical methods exist to evalu- continuous weak zone where all the subsequent plastic ate this critical length (Puzrin and Germanovich 2005; deformation occurs, generally known as a shear band Quinn et al. 2012; Zhang et al. 2015). Propagation of (Zhang and Wang 2020). The energy released during shear bands beyond their critical length leads to large strain softening acts as the driving force for the propa- retrogressive failures in sensitive clays. Detailed descrip- gation of the shear band. The shear band propagates tions of the formation of the sliding surface in sensitive steadily as long as the mobilized strength within the clay landslides can be found in Locat et al.’s (2011) work. shear band is lower than the peak strength but greater than the remolded strength. When plastic deforma- Landslide types and failure mechanisms tions further reduce the shear strength in the weak zone Sensitive clays (quick clays) are postglacial marine to the remolded strength, the shear band can progres- deposits found primarily in North America (East- sively propagate without additional external load (Quinn ern Canada and Alaska) and Scandinavia (Torrance et al. 2011). The required length for the catastrophic 1983). The largest deposits of postglacial marine clays Shear Stress (kPa) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 6 of 28 Fig. 4 Change in undrained shear strength ratio with strain rate (after Lefebvre and LeBoeuf 1987) were formed in the Champlain Sea in the Saint-Law- Different authors tried to classify landslides in the pres - rence lowland, approximately 12,500 to 10,000 years ence of sensitive clays. Varnes (1978) identified two kinds BP (Hillaire-Marcel 1980). Lefebvre (1986) identified of landslide, "spreading by lateral failure" and "earth flow" landslides as an important feature of valley formation if the mass slides or flows, respectively. Carson and Lajoie caused by erosion of rivers and streams. He emphasized (1981) classified the landslides in the sensitive marine that the erodibility of the clay layers and change in sediments into five categories: two-dimensional spread - groundwater regime in these deposits are major causes ing failures, aborted retrogression, excess retrogression, of instability. Demers et al. (2014) illustrated that the multidirectional retrogression, and flakeslides. The last majority of the landslides in Canadian sensitive clays four are comparable to the earth flows defined by Var - have occurred along the watercourses where erosion nes (1978). Tavenas (1984) and Karlsrud et al. (1984) acted as the main triggering factor. Quinn et al. (2011, classified the landslides observed in the sensitive clays 2012) stated that the development of a failure surface of Canada and Scandinavia based on the type of move- in a sensitive clay landslide might be very slow (years ment involved in the slides as single rotational slides, of small erosions) or rapid (earthquake, pile driving, multiple retrogressive slides (earth flows/flow slides), blasting, or other sudden shocks) depending on the translational progressive flake slides, and spreads. Locat triggering factor. However, if a slope has marginal sta- et al. (2011) identified the last three types to occur sud - bility, a small increase in stress in the slope (e.g., due to denly and affect large areas. Hungr et al. (2014) updated seasonal variation of pore water pressure) can cause a Varnes’s (1978) classification into 21 separate categories, catastrophic failure (Lefebvre 1996; Quinn et al. 2011; among which landslides observed in sensitive clays are Urmi et al. 2022). Various hydrological factors, includ- divided into sensitive clay flow-slides (multiple retrogres - ing precipitation, piezometric pressure, groundwater sive slides/earth flows) and sensitive clay spreads (trans - flow, and other processes, are recognized as contribu - lational progressive flake slides). The following two tors to triggering retrogressive failures in sensitive subsections present a literature review on the two most clays (Donovan 1978; Eden and Mitchell 1970; Lefebvre important mechanisms: flow slides and spreads in sensi - 1986). Case studies and discussions of large landslide tive clay. events in glaciomarine clay often identify hydrogeo- Although spreads and flow slides are well distin - logical factors, such as rainfall, snowmelt or anomalous guished in the literature, there is no clear agreement on weather leading up to the event, as a possible trigger how to differentiate their occurrence based on the mate - (Evans and Brooks 1994; Karrow 1972). rial properties and characteristics of the slope. Demers Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 7 of 28 et al. (2014) analyzed the characteristics of historically strength (Fig. 5c). When the soil mass becomes remolded, recorded flow slides (over 60) and spreads (about 40) it behaves like a viscous liquid (debris) and flows out of and showed similar material properties for both mecha- the newly formed crater with a considerable velocity. nisms. Some researchers pointed out that spreads can This movement and stress redistribution trigger a series occur in soils with low sensitivity where a flow slide will of retrogressive rotational failures (Fig. 5d). The rapid not occur (Carson 1977; Demers et al. 2014; Locat et al. removal of lateral support combined with the low per- 2008). Quinn et al. (2007) suggested that spread occurs meability of the sensitive clay generates negative excess where the thickness of the crust is larger compared to the pore water pressures in the back scarp (i.e., undrained sensitive clay layer; thus, the remolded clay flows under - conditions) increasing the effective stress (Mitchell and neath and squeezes up through the cracks. In the event Markell 1974). Therefore, each successive retrogression of flow slides, the upper crust is relatively thin compared must overcome a greater average shearing resistance due to the sensitive clay layer; hence the crust is carried away to the changing initial stress conditions as the back scarp with the remolded clay. Demers et al. (2014) found that becomes further removed from the initial slope. Eventu- spreads occur when the sensitive clay layers exist below ally, the back scarp remains stable, no more debris can the watercourse level, contrary to flow slides. flow out, and the sliding mass stabilizes (Fig. 5e). Mitchell and Markell (1974) estimated that the transi- Flow slides tion between drained and undrained conditions happens Flow slides initiate with a single rotational failure that is at a horizontal distance from the toe of the slope between usually a result of a slow decrease of the shear strength 3Hsecβ and 4Hsecβ (H being the slope height and β the in the sensitive clays due to leaching out of salt through slope angle) because the stresses within the slope drasti- pore water in the process of erosion over decades (i.e., cally changes from K condition at the mentioned hori- drained conditions) (Bjrrum 1955). Consistently, the zontal distance. Therefore, the authors suggested that, initial failure should be analyzed in drained conditions for a failure starting in drained condition, retrogression except for the cases when it is initiated by sudden loading beyond a horizontal distance of 3Hsecβ to 4Hsecβ from (e.g. earthquake or intense rainfall). In soils that are not the toe of the slope should be considered as an und- sensitive, the initially mobilized mass rapidly becomes rained or short-term failure. For the undrained analysis, stable (Fig. 5a, b). However, in sensitive clays, the shear a flow slide occurs if the stability number, N = γH/s is s up strength within the failure surface and the deformed larger than 6 (γ being the soil’s natural unit weight and mass reduces dramatically down to the remolded shear s the undrained shear strength), and it terminates due up Fig. 5 a–e Flow slide mechanism (after the description of Bjrrum 1955) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 8 of 28 to topographic/stratigraphic restrictions or when the retrogressive failure. Demers et al. (2014) showed by energy dissipated in the flowing debris becomes equal to analyzing the previously occurred flow slides that most the energy released at the back scarp. sites had very low remolded shear strength (< 0.8 kPa) Other studies on flow slides in sensitive clays focus and a very high liquidity index (1.5–16). To summarize, on the criteria for the occurrence of large retrogres- the aforementioned researchers acknowledged that large sive landslides. Lebuis and Rissman (1979) stated that flow slides are most likely to occur when the following large retrogressive landslides occur if the liquidity index conditions are met: S > 25, N > 4, w < 40%, I > 1.2, and t s L L (I ) is greater than 1.2 or the remolded shear strength s < 1 kPa. L ur (s ) is less than 1 kPa. Tavenas et al. (1983) argued that ur the condition based on mechanical soil parameters is Spread necessary for assessing retrogression potential but is Spreads are identified by their unique structure of alter - inadequate. One must account for whether the energy native crests and level surfaces after failure, generally dissipated by the initial slide is large enough to remold known as horst and grabens (Fig. 6). The mechanism of the clay so that it can flow out of the crater. They summa - spread was first described as a retrogressive failure by rized all the conditions to assess retrogression potential Odenstad (1951) for the Skottorp landslide in the Lidan in four points. Firstly, an initial slope failure must occur river. Odenstad (1951) stated that the slide started with against long-term stability. Secondly, the back-scarp a part of the riverbank slipping into the river. The driv - failure in undrained conditions will occur only if N > 4. ing factor for the failure initiation was not distinctly iden- Thirdly, the deformed soil mass from the first slide will tified, but erosion, leaching of salt, or blasting activity get remolded if the remolding index (I ) > 70% or liquid near the river could be responsible for the slow reduc- limit (w ) < 40%. Finally, the remolded debris will flow if tion of the slope stability over time. The formation pro - I > 2 or s < 1 kPa. Lebuis et al. (1983) also stressed that cess of horst and grabens is depicted in Fig. 6. When a L ur the energy required for the clay to be wholly remolded slope experiences instability, the strength of the weak should be dissipated from the first slide. They added soil layer drops due to stress concentration within this another observation from case studies that earth flows layer and failure propagates horizontally through the with retrogression of 100 m occurs when the sensitivity layer opposite to the river producing a slide bottom "ba", (S ) is greater than 25 and remolded shear strength (s ) simultaneously, a slip surface "bc" is formed at an angle t ur is less than 1 kPa. Later studies somewhat agree with the of 45° parallel to the riverbank (Fig. 6a). When stresses previous criteria for large retrogressive failure. along "bc" reach near to zero it starts to slip forming a For example, Lefebvre (1996) stated that slopes with wedge "bb’c’c" (Fig. 6b). When "bb’c’c" slips from posi- Ns > 6 and s < 1.5 kPa have the risk of retrogressive fail- tion 1 to 2, it stops at a depth where it reaches the slide ur ure after an initial slide; Thakur and Degago (2012) stated bottom forming a graben (Fig. 6c). The drag of this slip that soils having s > 1 kPa are less likely to initiate large causes a rupture "de" parallel to "bc". By this time, the ur Fig. 6 a–f Spread mechanism (after Odenstad 1951) Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 9 of 28 slide bottom has already reached point "d" (Fig. 6c). The spreads laterally like a liquid (Fig. 7a–c). The remolded horizontal drag and vertical slip cause a secondary failure clay is squeezed up through the cracks towards the top surface along "cd" creating a wedge "cde" (Fig. 6d). The layer, and the top layer is stretched and separated into wedge "cde" sinks from position 3 to 4, creating a graben segmented blocks (Fig. 7b), which might have some which leaves a horst at "bcd" (Fig. 6e). This is how the rotational movements due to frictional drag. This pro - whole strip "SS’" forms in a retrogressive and discontinu- duces ridge or rib-like patterns (Fig. 7c). The failure ous manner (Fig. 6f ) and finally slips into the river with stops with the increased frictional resistance of the an almost horizontal (slightly inclined towards the bot- squeezed-up liquid and increasing strength due to pore tom) translatory movement. pressure dissipation in the soil (Fig. 7d). Mollard and Mollard and Hughes (1973) argued that the landslides Hugh’s explanation lacks the detailing of the influence in the Grondines and Trois Rivieres Areas, Quebec, of the tension cracks to create the prism-like structures explained by Karrow (1972) as multiple rotational retro- in a spread. gressive slides are instead a spreading failure. They rea - Carson (1977) modified Odensta’s (1951) spread mech - soned that earthflow cannot explain the formation of anism and described the development of tension cracks parallel ridges. Their description agrees with Odenstad in addition to horizontal subsidence (Fig. 8). He ana- (1951) that the trigger for the initial instability could be lytically explained the mechanism of the squeezed-up any of the natural or anthropogenic reasons and that remolded clay in tension cracks aiding the formation of there exists a weak sensitive clay layer below the ground horst and grabens, combining both descriptions above. surface. He postulated that the width of the cracks filled up with Unlike Odenstad’s horizontal propagation of the slide remolded clay at the end of the landslide equals the total bottom, they stated that with appreciable stress con- volume of remolded clay in the weak layer. He empha- centration, the sensitive clay layer gets remolded and sized that spread could occur regardless of the sensitivity Fig. 7 a–d Mechanism of spread (after Mollard and Hughs 1973) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 10 of 28 Fig. 8 Spread mechanism (after Carson 1977) Fig. 10 Spread mechanism (after Locat et al. 2011) movement accelerates the translational slide. Finally, the rapid progression of the soil mass above the liquified clay results in horst and grabens (Fig. 10). Numerical analysis of sensitive clay landslides Fig. 9 Saint-Ligouri landslide (after Leroueil et al. 2011) Numerical modeling of sensitive clay landslides should address three particular challenges: strain-softening, remolding (solid to liquid transition), and large deforma- being high or low; the important factor is how rapidly the tions. The following subsections present and discuss the soil gets disturbed to generate a rapid flow. prospect of the constitutive soil models and numerical Grondin and Demers (1996) suggested that the forma- frameworks considered in the literature to model sensi- tion of horst and grabens may not be discontinuous as tive clay landslides. the description by Odenstad (1951) and Carson (1977) because, in the landslide of St-Ligouri, the top surface of the grabens was connected by grass (Fig. 9). They sug - Constitutive soil models for the study of sensitive clays gested the landslide occurred due to horizontal subsid- The constitutive soil model, which represents the stress– ence that caused the dislocation of the upper soil mass. strain behavior of the soil, is the most critical compo- A well-described discussion on the mechanism of nent for modeling sensitive clay slopes. The capability to spread can also be found in Locat et al. (2011). They reproduce realistic strain-softening characteristics in the described this mechanism as progressive (upward pro- material model is necessary for more accurate numerical gressive failure) rather than retrogressive (Locat 2007; analyses of large deformation problems in such materi- Quinn et al. 2007). As per their hypothesis, a failure als. The constitutive models used in previous works for surface in the sensitive clay layer, almost horizontal to modeling sensitive clay landslides are summarized below. the ground, moving upward starting near the toe of the Their ability to deal with (a) strain-softening behavior, slope, is formed before any noticeable movement occurs. (b) strain-rate effects, and (c) rheological behavior of The failure initiates with a slow translational movement remolded soil is discussed. For completion, stress–strain of the soil mass above the failure surface. Then, the sen - curves numerically predicted by some of the models are sitive clay along the failure surface becomes remolded compared with DSS data from sensitive clays. In particu- and liquefies due to the accumulated deformation. The lar, those constitutive models that have been used in the Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 11 of 28 literature to reproduce real case scenarios with available 2016b) used this model to simulate retrogressive failure stress–strain DSS curves such data are considered for features in sensitive clays together with a strain-softening comparison (Locat et al. 2015; Tran and Sołowski 2019; law (Eq. 1), where the undrained shear strength linearly Zhang et al. 2020). Otherwise, the stress–strain curve reduces with deviatoric plastic shear strain ( ε ) from the of Saint-Barnabé-Nord (Locat 2007) is considered for peak (s ) to a residual value (s ) (Fig. 11a). up ur reference. s + Hε ; ε < ε up p p pr s ε = u p (1) s ; ε > ε ur p pr Von‑Mises based models The Von-Mises model is one of the simplest elastoplastic where ε is the plastic deviatoric strain at the onset of pr models to simulate the undrained behavior of clay. The the remolded strength, and H is the linear softening model generally considers associated plastic flow and, at coefficient. failure, it does not allow for volumetric strain. It requires The capacity of this model to reproduce the stress– only two parameters: the undrained elastic modulus strain curve of Saint-Barnabé-Nord (Locat 2007) (E ) and the undrained shear strength (s ). The Poisson’s is presented in Fig. 12b. For the determination of u u ratio is assumed constant (ν = 0.5). Wang et al. (2016a, the softening modulus, the value of the strain at (a) (b) Saint-Barnabé-Nord 23.31m Remolded strain 60% Remolded strain 100% Remolded strain 150% Extrapolation of the DSS curve 050 100 150 200 Shear Strain (%) Fig. 11 a Stress–strain behavior in a cohesion softening model ( Wang et al. 2016b, 2016a) b comparison between Wang et al.’s (2016b, 2016a) model and Saint-Barnabé-Nord clay Fig. 12 Stress–strain behavior of sensitive clay (Dey et al. 2015) Shear Stress (kPa) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 12 of 28 remolded shear strength is required. Based on Thakur The non-linear part of Eq. (2) (curve "bcd" in Fig. 12) and Degago (2014), the strain at remolded shear is a modified form of the strength degradation equation strength is estimated to be 300% for Norwegian sen- proposed by Einav and Randolph (2005) as follows, sitive clays. Assuming the strain at remolded shear 3γ s s s u ur ur strength as 60–150% and considering s = 79.5kPaand up 95 = + 1 − × e (3) s s s up up up s = 1.6kPa based on the experimental results from ur Locat et al. (2008), the value H varies between −135 where displacements relate to the strain as δ=γ t (t = shear to −52 kPa. It is observed from Fig. 11b that the lin- band thickness). ear approximation either overpredicts or underpredicts The linear-elastic pre-peak segment (line "oa" in Fig. 12) the actual curve obtained from laboratory testing. Even is defined using undrained stiffness parameters. The peak though a linear approximation cannot capture the non- undrained shear strength (s ) is mobilized at point "a" up linearities of the softening behavior of sensitive clays, and remains constant up to point "b" for a displacement a reasonable approximation can be achieved when the of δ from point "a". The rest of the curve corresponds pc areas between the stress–strain curves produced by the to Eq. (2). constitutive model and the actual stress–strain curve The comparison between the post-peak stress–strain compensate (e.g., the case with remolded strain taken relationship predicted by Eq. (3) and the DSS stress– as 100%). strain curve of Saint-Barnabé-Nord (Locat 2007) is pre- A group of researchers (Dey et al. 2016a, 2016b, 2015; sented in Fig. 13. The comparison has been made in Islam et al. 2019) used the same yield criterion with a terms of shear strain instead of displacement. After a non-linear strength degradation as a function of plastic calibration process, the most fitting curve is obtained shear displacement (Fig. 12) as follows, with γ =130%. The predicted curve shows some discrep - � � ′ 3δ ancies with the DSS data up to 20% strain, but it exactly s s uR uR ′ + 1 − e if 0 ≤ δ < 2δ s s up up matches from 20–46% strain. s s −s δ −2δ = uR uR ur 95 ′ − if 2δ ≤ δ <δ It can be concluded that exponential strain softening 95 r s s s δ −2δ up up r 95 up s ′ ur is better suited than linear strain-softening laws to cap- if δ ≥ δ up ture the strain-softening from sensitive clays, but much (2) uncertainty prevails in the accurate estimation of the where s is the mobilized undrained shear strength; parameters like γ and γ . The wrong approximation of r 95 δ = δ − δ + δ with δ and δ being the elastic and t e pc e t these parameters can significantly reduce the accuracy of total shear displacements, respectively; δ is the value the strain-softening equation for field conditions. of δ at which the undrained shear strength is reduced Wang et al. (2021, 2022) added strain-rate effects on by 95% of (s − s). s is the degraded strength at a up uR uR the shear strength using Eq. (4). They considered the displacement of 2δ after the peak.δ is the value of δ 95 r undrained shear strength ( s ) as a function of a shear remolded shear strength (s ). ur strain-softening factor ( f ) and a strain rate factor ( f ) a s 1 2 Saint-Barnabé-Nord 23.31m Strain at 95% strength decrease=100% Strain at 95% strength decrease=130% Strain at 95% strength decrease=160% ------- Extrapolationofthe DSScurve 050 100 150 200 Shear Strain (%) Fig. 13 Comparison between the exponential strain-softening law with different γ to the stress–strain behavior from DSS test from Saint-Barnabé-Nord clay Shear Stress (kPa) Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 13 of 28 (Eq. 6) but also incorporates the viscosity and shear thin- s = s f f (4) u uy 1 2 ning index like Herschel–Bulkley model. This unified where s is the undrained shear strength at a very low equation is termed an "additive power law" model and uy strain rate (i.e., quasi-static). expressed as, Generally, the dependency of undrained shear strength γ˙ of clays on applied strain rate has customarily been char- s = 1 + η s u u,ref (8) γ˙ ref acterized in terms of a semi-logarithmic relation (Gra- ham et al. 1983; Zhou and Randolph 2009) (Eq. 5) and a Wang et al. (2021, 2022) used Eq. (8) in his constitutive power law (Eq. 6) (Einav and Randolph 2005) expressed model to incorporate strain rate, and the second function as, in Eq. (4) is defined as, γ˙ s = 1 + µlog s u γ˙ u,ref (5) γ˙ f = 1 + η ref 2 (9) γ˙ ref By incorporating strain rates, constitutive models can γ˙ s = s u (6) u,ref effectively evaluate the impact of a time-dependent strain γ˙ ref or creep behavior on both the onset and advancement of failure. where γ ˙ is the shear strain rate, γ ˙ is the strain rate at ref a reference shear strength s , the coefficient μ and β u,ref give the proportional change in shear strength for each Tresca based models order of magnitude change in strain rate, which lies in the Tresca is also an elastoplastic model, very similar to the range of 0.05–0.2 and 0.05–0.1, respectively. Von-Mises model. Its simplicity inspired some research- Again, the behavior of liquified remolded sensitive ers to model sensitive clays by adding strain-softening clays in debris flow can be described with a strain rate laws (Shan et al. 2021; Tran and Sołowski 2019; Yuan dependent fluid mechanics framework (Herschel–Bulk - et al. 2020; Zhang et al. 2015). Locat et al. (2013) mod- ley model) relating the yield stress (τ ) with strain rate ( γ ˙ ) eled the spread mechanism in sensitive clays assuming a based on fluid viscosity ( η ) and a shear thinning index (n) hyperbolic elasticity model up to the peak stress and lin- as (Deglo De Besses et al. 2003), ear plasticity in post-peak strain softening (Fig. 14a). The strain-softening is linear and defined with a softening γ˙ = 0 for |τ| ≤ τ modulus K as follows, n (7) τ = τ + η|γ˙| for |τ| >τ 0 0 s − s up ur K = (10) For the sensitive clays, the effect of strain rate on the δ − δ r p undrained shear strength would be present in the solid phase as well as in the liquid phase when it flows like where δ and δ are the shear displacement at peak and p r debris. To capture the strain rate effect on strength in remolded strength, respectively. both the phases Zhu and Randolph (2011) proposed This model has also been applied to study the post- a unified equation that originates from the power-law peak behavior of the Sainte-Monique landslide in Fig. 14 a Shear stress-displacement relationship (Locat et al. 2013), b comparison of assumed strain-softening with the laboratory test result Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 14 of 28 γ γ˙ Quebec (Locat et al. 2015), assuming γ = 2% and s = max s − s − s , s 1 + K u up up ur ur γ = 55% . The comparison between the linear approx - γ γ˙ r ref imation and the measured stress–strain curve is (13) presented in Fig. 14b. It is evident that linear approxi- where n is the power index (usually varies from 0.4 to 0.6 mations significantly deviate from the actual stress– for Canadian sensitive clay), and K is the viscosity coef- strain curves. 0.28 ficient ( K ≈ 0.028s ). The depth-wise variation of peak up Zhang et al. (2015) investigated the initiation and strength was defined as follows: propagation of a fully softened shear zone in subma- rine sensitive clay landslides with both linear (Eq. 11) s 2m ≤ h < 5m up1 s = (14) up s + a(h − 5) h ≥ 5m and non-linear (Eq. 12) strain-softening laws. The lat - up1 est (Eq. 12) is based on Einav and Randolph (2005) where h is the soil depth, s is the undrained shear up1 (Fig. 15). strength in the first clay layer ( h =2 m-5 m), and " a " is They compared the critical length of the softened the strength gradient along the depth h in the second clay zone predicted by the numerical model with the analyt- layer ( h >5 m). ical evaluation and concluded that linear degradation Tran and Sołowski (2019) proposed a modified version of overestimates the length of the shear band by 10–15% the elastoplastic Tresca model with a non-associated flow more than non-linear degradation. The softening equa - rule for modeling the progressive failure behavior of sensi- tions are as follows, tive clays and used it to simulate the Sainte-Monique land- slide. They added features to the model for replicating the s = s − s − s u up up ur (11) field soil behavior, including strain-rate effects, the effect of water content, and soil depth on the shear strength. The non-linear strain-softening law describes the strength deg- 95 (12) s = s + s − s e u ur up ur radation as follows, where s is the degraded undrained shear strength at γ˙ 1 1 −3γ /γ displacement increment γ after reaching the peak, and γ 95 s = s + 1 − e when γ >γ u u,ref e γ˙ S S ref t t is the plastic strain increment required to reduce the (15) strength by 95% of (s − s ) from the peak strength ( s up ur up ), where γ is the accumulated elastic strain. The reference Shan et al. (2021) added strain-rate effects and depth- undrained shear strength depends on the water content wise variation on shear strength to Eq. (11) as ( w ) and depth ( h ) a s : Fig. 15 Stress–strain behavior with exponential strain-softening (Zhang et al. 2015) Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 15 of 28 60 50 Saint-Monique 13.45m Sainte-Monique 13.45m Tran and Solowski 2019 Jin et al. (2020) η=1 Jin et al. (2020) η=2 Jin et al. (2020) η=3 020406080 100 Shear Strain (%) Fig. 16 Comparison of stress–strain behavior of DSS test result with Shear Strain (%) the numerical model of Tran and Solwoski (2019) Fig. 17 Comparison between DSS test results vs. stress–strain behavior of Jin et al. (2020) with different η values −b s (w, h) = a w s + �s h − h 1 u u,ref uh,ref ref plastic viscosity (η ). As shown in Fig. 18, the fluid starts (16) to flow after reaching its yield stress, and the yield stress where, a is the undrained shear strength at w = 100% at increases linearly with the shear rate. Several rheologi- a reference strain rate ( γ ), w (%) is the water content cal models have been used for simulating the debris flow ref and b is the model parameter, s is the reference und- of sensitive clays, including the Herschel–Bulkley model 1 uh,ref rained shear strength at the reference depth h , and s (Imran et al. 2001; Turmel et al. 2020; Turmel and Locat ref u is the increase of strength per unit depth after the refer- 2018). The Bingham model is a limiting case of Herchel– ence depth. The stress–strain prediction of this model Bulkley rheology. It should be noted that these models is compared with laboratory DSS test result (Sainte- only consider the liquid flow of the sensitive clays and do Monique landslide), which shows that this model is bet- not capture the transition from solid to a liquid phase. ter suited compared to previous linear models discussed With the idea of capturing the transition from solid above in terms of capturing the non-linear stress–strain to liquid behavior of sensitive clay, Zhang et al. (2017) behavior of sensitive clays (Fig. 16). came up with an elastoviscoplastic model that com- Jin et al. (2020) simulated retrogressive failure in sensi- bines an elastoplastic model with a viscous model using tive clays with a cohesion softening model which is com- the concept of strain-based transition initially proposed parable to the Tresca softening model where the peak by Prime et al. (2014). In particular, they combined the and residual cohesion (c and c ) are equivalent to the Bingham model and the Tresca model with linear strain p r peak and remolded undrained shear strengths ( s and softening (Zhang et al. 2018, 2020, 2017). Total strain up s ) of sensitive clays. The softening is defined as, rate ( ε ˙ ) for the elastoviscoplastic material is defined as ur the summation of an elastic strain rate ( ε ˙ ) and a visco- −ηε s = s + s − s e vp u ur up ur (17) plastic strain rate ( ε ˙ ). e vp where η is the shape factor that controls the rate of ε˙ =˙ε +˙ε (18) strength decrease. The strain is purely elastic when the stress state is They evaluated mesh dependency and the effect of the below the Tresca yield surface, whereas viscoplastic shape factor η on post-failure behavior of the landslides. strain starts to develop when the stress state crosses the It can be seen in Fig. 17 that, the softening curve is sig- yield surface. Strain softening is incorporated by reducing nificantly dependent on the value of the shape factor ( η ). For the case of Sainte-Monique landslide, η = 2 is more representative of the field behavior. Bingham‑Tresca based models The constitutive models described above are common in the study of soil mechanics, and most of them do not directly consider the rheological properties of the clay, which may affect the prediction of runout distances in retrogressive flow slides. In the field of fluid mechan - ics, the Bingham plastic model is a common constitu- tive law that considers non-Newtonian rheology to predict the flow behavior using its yield stress ( τ ) and Fig. 18 Bingham plastic model Shear Stress (kPa) Shear Stress (kPa) Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 16 of 28 modeling geomaterials, particularly for large deforma- the undrained shear strength s using a bilinear function tion problems, have been summarized by Augarde et al. (similar to Fig. 11a) of the equivalent deviatoric plastic (2021). Before that, Soga et al. (2016) and Wang et al. strain. In this case, the softening modulus H is a function vp (2015) compared the advantages and disadvantages of of the viscoplastic strain ( ε ). existing numerical frameworks used in slope stability The classical Bingham model is utilized to describe the analysis and other geotechnical problems, respectively. rheological behavior of sensitive clays, and the total stress In the following section, large deformation numerical is defined as, frameworks that have particularly been used in the analy- vp σ = τ + ηε˙ (19) sis of landslides with strain-softening materials or sensi- tive clays will be discussed, which are mainly different where τ is the stress lying on the boundary of the Tresca formulations in continuum approaches. vp yield surface, η is the viscosity coefficient and ε ˙ is the viscoplastic strain. When the viscosity is zero, the model Updated Lagrangian finite element method reduces to the classic Tresca model. This model is com - The most popular approach used in geotechnics is parable to the strain-rate-based models, where the strain- the standard finite element method (FEM) with clas - rate dependency of liquified debris is derived from the sic Lagrangian formulation (known as the total Lagran- Herschel–Bulkley model (Eqs. 7–9). Zhang et al. (2019) gian, TL). The TL is a continuum method that uses the used this constitutive soil model to simulate the Saint- undeformed initial geometry as its frame of reference Jude landslide (2010) and showed that ignoring rheo- for computing the static and kinematic variables as well logical properties overestimates the runout distance. as formulating the discrete equations. In the cases where However, the linear strain-softening is not representative deformations are substantial with respect to the initial of the actual stress–strain behavior (Fig. 19). geometry, this formulation suffers from mesh distortion and produces inaccurate results. The updated Lagrangian Numerical frameworks for modeling landslides in sensitive (UL) formulation is an upgraded version of the classic clays lagrangian formulation to solve the mesh distortion prob- Numerical analysis has two prominent approaches, i.e. lem for its implementation in large deformation problems continuum mechanics and discrete mechanics. In con- (Bathe 1996). In UL formulation, the variables are com- tinuum analysis, each particle within a material is not puted, and equations are formulated for the deformed treated explicitly, and the material is modeled as a con- state in the previous calculation step. Therefore, the posi - tinuous entity; the change in the properties of the geo- tions of the nodes are updated based on the displacement material under loading conditions is represented by a calculated in the last increment. The UL framework is constitutive model. On the other hand, discontinuum available in some commercial FEM software, making methods model the geomaterial as a collection of distinct it flexible to work with. The limitation of this method is particles which may or may not represent a real particle that when very large deformation is encountered, ele- arrangement, and the particles interact through their ments become too distorted. This distortion reduces the contacts with other particles and boundaries. Numeri- accuracy of the results and creates computational insta- cal analysis for large deformation problems in geome- bilities that make the calculation impossible (Augarde chanics has been a prominent issue in recent times. Both et al. 2021). Mohammadi and Taiebat (2013) have used continuum and discontinuum numerical frameworks for the UL FEM formulation to simulate progressive failure in strain-softening soil slopes. In their model, mesh dis- tortion was encountered for higher sensitivity (RF = 1/S ), as shown in Fig. 20. Saint-Jude 22.7m Zhang et al. (2019) ----- Extrapolationofthe DSScurve Arbitrary Lagrangian–Eulerian (ALE) methods ALE is the upgraded version of UL, which offers a solu - tion to the mesh distortion by substituting the distorted mesh with a new mesh (re-meshing). This requires the transfer of state variables (re-mapping), which might reduce the accuracy of the solution when these are 04590135 180 history-dependent. This method is referred to as arbi - Shear Strain (%) trary Lagrangian–Eulerian because the process of mate- Fig. 19 Comparison between DSS test results vs. stress–strain rial state transformation from the old mesh to the new behavior of Zhang et al. (2017) one is similar to that of the Eulerian description. Several Shear Stress (kPa) Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 17 of 28 Fig. 20 The deformed FEM mesh for different RF values with UL (after Mohammadi and Taiebat 2013) re-meshing and re-mapping methods have produced sev- deform freely. The two domains interact with each other eral branches of the ALE formulation. The Re-meshing through some contact algorithms. The CEL approach and Interpolation Technique combined with Small Strain is also available in commercial software like ABACUS. (RITSS) approach is a widely popular re-meshing tech- Dey et al. (2013, 2015, 2016a, 2016b) used this frame- nique. The RITSS approach addresses large strain prob - work to model progressive failures leading to spreads in lems by dividing them into smaller Lagrangian strain submarine sensitive clay slopes. Islam et al. (2019) used increments (Hu and Randolph 1996). At the end of each this framework to model retrogressive failure initiated increment, the deformed body is re-meshed with new by earthquake loads. Wang et al. (2021, 2022) modeled undistorted elements. To ensure an accurate transfer of retrogressive failures resulting from erosion with some information, solution variables such as stresses, defor- practical applications using this method. The authors mations, velocities, nodal pore pressures, etc., are then adapted the techniques to reduce the mesh dependency interpolated from the old mesh to the new mesh (Ullah of the models. et al. 2018). The RITSS process can be broken down into four main steps, which include the generation of an ini- Particle finite element method (PFEM) and smoothed particle tial mesh, conducting an incremental step of the Lagran- finite element method (SPFEM) gian analysis, updating boundaries and re-meshing, The Particle Finite Element Method (PFEM) is also an and mapping stresses and material properties from the approach mainly developed to overcome the mesh distor- old to the new mesh. These steps are repeated until the tion suffered by FEM. In this method, the mesh nodes are entire large deformation analysis is completed using a regarded as particles that can freely move even beyond standard Lagrangian finite element package (Tian et al. the initial computational domain. The particles carry all 2014). This process, thus, overcomes mesh distortion and information. The boundary of the domain is marked by allows for the modeling of various geotechnical prob- connecting the outermost particles, and each internal lems, including those with fully drained, undrained, or particle is connected through the Delaunay triangulation intermediate drainage conditions. It’s important to note technique. After the mesh is established, the governing that the accuracy and success of this process depend on equations are solved as it is in a conventional FEM simu- the choice of interpolation scheme and mesh density. lation. The updated positions of the mesh nodes are used The RITSS approach can be flexibly used in any FEM to form a new cloud of particles, and the process repeats code but is computationally very expensive. Zhang et al. (Fig. 21). (2015) used the RITSS approach to simulate the initiation The main advantage of this method is that it shows and propagation of a shear band in a fully softened weak convergence behavior regardless of a large change in the zone. Zhang et al. (2019) simulated a landslide in subma- state variables from the previous calculation step to the rine sensitive clays with the same framework. Shan et al. current step, which is very useful for modeling history- (2021) simulated the Sainte-Monique landslide in Que- dependent materials like sensitive clay landslides (Zhang bec in 1994 with RITSS and successfully simulated some et al. 2018, 2017). Zhang et al. (2020) utilized this frame- features of the retrogressive failure. work to simulate the Saint-Jude landslide in Quebec, One of the most popular versions of ALE is the Cou- Canada. It has been able to reproduce the failure mode pled Eulerian–Lagrangian (CEL) method, also known and the progressive failure process of the Saint-Jude land- as multi-material ALE. This method utilizes the advan - slide and correctly estimated its final runout distance, ret - tages of both the Lagrangian and Eulerian approaches rogression distance, and failure surface. The infinitesimal (Qiu et al. 2011). This method has two different domains strain assumption in this method is likely to result in sev- for material description. One is the classic Eulerian, and eral errors, including excess strain for rigid body motion. the other is the updated Lagrangian. Stiffer materials are Zhang et al. (2017) reported that the error is limited to described with the Lagrangian domain. When stresses 1% for a converged solution. Contact between solid–solid are applied on these stiffer materials, it causes the softer and solid–liquid is complex in this method. Frequent re- materials in the vicinity to experience large deformation. meshing in large deformation problems requires extreme The Eulerian description allows these softer materials to precision in transferring history-dependent information Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 18 of 28 Fig. 21 Steps for the PFEM modeling of a landslide from the old mesh to the new one to maintain the accu- to the next step. The background mesh must contain the racy of the calculation (Augarde et al. 2021). problem domain geometry, and the material deformation The Smoothed Particle Finite Element Method is carried by the material points. Thus, the mesh remains (SPFEM) is an improved form of PFEM where the equi- undistorted throughout the calculation, and mesh distor- librium of the governing equations is established at the tion is completely avoided. Some numerical difficulties nodes instead of the Gaussian points with the help of of MPM are cell crossing instability, generation of non- strain smoothing cells covering each node. This reduces physical stiffness, and the mapping from material points the error in re-mapping the history-dependent variables. to nodes and back. This method is more accurate and computationally flex - The full process of retrogressive failure of sensitive ible than PFEM (Zhang et al. 2018). With this framework, clay triggered by erosion has been simulated using Yuan et al. (2020) simulated retrogressive failure in sensi- MPM framework by Wang et al. (2016b, 2016a) and tive clays. This simulation used the simplest form of the Tran and Sołowski (2019). The simulations well cap - strain-softening Tresca model. Therefore, the applicabil - tured the features of a retrogressive landslide. ity of this framework with a more sophisticated consti- tutive model with rate effects and non-linear softening is yet to be studied. Material point method (MPM) MPM is a combination of particle-based and mesh-based methods. The computational domain is discretized with a number of particles (i.e. material points) and a fixed background mesh. The material points are described in the Lagrangian formulation. Each material point rep- resents a part of the computational domain and moves through the fixed background mesh. All the physical properties of the continuum are carried by the material points. The background mesh is described in Eulerian formulation. The state variables are transferred from the material points to the mesh nodes in each computational step (Fig. 21: Steps for the PFEM modeling of a landslide Fig. 22), and the balance equations are solved at the nodes of the Eulerian mesh. Updated properties are mapped Fig. 22 MPM Computational cycle (Fern et al. 2019) back to the material points, and the calculation proceeds Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 19 of 28 Contribution of numerical analysis to understand to a great extent. For a varying slope angle (β) of a hori- the control parameters of sensitive clay landslides zontal soil layer (Fig. 23a), steeper slopes are more sus- Various combinations of the constitutive models and ceptible to larger retrogressive failure (Dey et al. 2016a, numerical frameworks described in the previous sec- 2016b; Islam et al. 2019; Locat et al. 2013). Jin et al. 2021 tion have been implemented to model different land - showed that higher β value reduces the time to reach fail- slides in sensitive clays. Some of these works explore the ure for sensitive clay slopes. For a varying inclination of failure mechanisms in detail and provide new insights. the clay layer α with the horizontal (Fig. 23b), a higher Numerical simulations also help understand which geo- α changes the slide movement from rotational to trans- metrical and material parameters control the occurrence lational and produces a higher number of secondary slip and post-failure behavior of sensitive clay landslides. The planes with smaller failure blocks (Wang et al. 2016a, b). following section focuses on how the numerical results Islam et al. (2019) further demonstrated with a varying aided in understanding the critical factors for sensitive upslope inclination (γ) (Fig. 23c) that flow slide occurs for clay landslides. low and high values of γ. In contrast, the failure pattern for intermediate γ is spread. Overall, these numerical Eec ff t of the geometry of the slope on the occurrence results align with the literature that the susceptibility and and post‑failure behavior extent of progressive failure are high on steep slopes (Lo Several numerical studies have evaluated how the geo- and Lee 1973). metrical variations of the soil layer or the slope change the failure pattern, retrogression, and runout distances. Eec ff t of material properties on the occurrence In particular, the effect of the thickness of the sensitive and post‑failure behavior clay layer, height and width of the slope, slope angle, and The effect of different material parameters on the initia - inclination of the soil layer are some aspects that have tion and the extent (retrogression and runout distance) been assessed. of the sensitive clay landslides has been assessed in light Dey et al. (2016b) showed that a minimum thickness of different numerical simulations. Dey et al. (2015) of the sensitive clay layer is required to initiate retro- showed that progressive failure would not occur for gressive/progressive failure, while Zhang et al. (2020) very low sensitivity (S < 3), spread occurs with medium concluded that an increase in the thickness of the sen-sensitivity (S ~ 5–7), and the failure pattern changes to sitive clay layer (H ), keeping the crust thickness con- a flow slide with higher sensitivity (S ≥ 10). Moreover, St stant, increases retrogression distance. Additionally, an they found that decrease in the shear strength of the increase in H changes the failure pattern from spread to crust changes the failure pattern from spread to flow St flow slide (Dey et al. 2015; Islam et al. 2019). Studies show slides. In contrast, Islam et al. (2019) illustrated that that for a fixed slope height, an increase in slope width the failure pattern changes from a flow slide to a com - increases the retrogression and runout up to an optimum bined spread and flow slide type with increased sen - value. After this point, further width increase results in sitivity. Even though both Dey et al. (2015) and Islam no significant change in post-failure behavior; that is, et al. (2019) used the same softening constitutive model slope width has an optimum value for the occurrence and numerical framework, the differences in their con - of large retrogressive failure (Yuan et al. 2020). The fric - clusions might be due to the different triggering factors tional coefficient of the base layer also controls the retro - considered in their analysis, i.e., toe erosion (Dey et al. gression. It is observed that the larger the basal friction, 2015) and earthquake (Islam et al. 2019). In any case, the less susceptible the slope is to retrogressive failure both works coincide in that higher sensitivity caused (Wang et al. 2016a, b; Yuan et al. 2020). The slope angle higher retrogression and runout distances. Zhang et al. controls the failure mechanism and post-failure behavior (2018) illustrated that in addition to the increased Fig. 23 Different types of slope inclination Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 20 of 28 retrogression and runout, it takes a longer time for the The viscosity of remolded clay also affects the post- sliding front to stop entirely with increased sensitivity. failure behavior. An increase in viscosity leads to a Yuan et al. (2020) added that the failed soil mass would decrease in both the runout distance and the retrogres- reach a longer runout distance with higher sensitivity, sion distance of a retrogressive failure because highly owing to a smaller dissipation of potential energy for viscous fluid would consume more energy to flow (Yuan remolding, leading to higher kinetic energy. et al. 2020; Zhang et al. 2018, 2017). Zhang et al. (2018) Tran and Sołowski (2019) showed that large retro- reported that the effect of viscosity is more significant gressive landslides are also dependent on low remolded for the soils with higher sensitivity and low remolded strength ( s < 2 kPa) with high sensitivity (S > 25). Sev- strength. Their simulation demonstrated that the unac - ur eral researchers demonstrated that the retrogression counted viscosity overestimates the runout distance by distance is significantly controlled by the remolded 35% and the retrogression distance by 20%. Mobility of shear strength. Increased remolded strength s (keep- the liquified debris also affects the failure mechanism. If ur ing all other material and geometrical parameters the liquified debris moves out of the crater easily once fixed) reduces the retrogression distance, and a further the retrogression has started, a flow slide is more likely increase in s completely stops the retrogression (Islam to occur; on the contrary, if the liquified debris has slower ur et al. 2019; Locat et al. 2013; Tran and Sołowski 2019; movement, the failure may result in spread (Wang et al. Wang et al. 2016a, b). Likewise, when the remolded 2022). strength approaches zero, there is a sharp increase in Tran and Solwaski (2019) demonstrated that over- retrogression and runout (Zhang et al. 2018). It can also looking the impact of strain rate on shear strength may be said that highly brittle soils are more susceptible to overestimate the retrogression distance. Wang et al. progressive failure. Brittleness of soil is generally quan- (2022) provided further clarity on this topic, explaining tified by the brittleness index (I ) defined as (Bishop the effect of strain rate during failure with respect to the 1971) reference strain rate in the constitutive model. They con - cluded if the strain rate at the time of failure exceeds the s s up− ur I = reference strain rate, it may overestimate the post-failure (20) up movement. Conversely, if the strain rate is lower than the reference strain rate, it may underestimate of the final If retrogressive failure initiates, increased brittleness retrogression and runout distance. of the soil changes the failure type from spread to flow Wang et al. (2016a) illustrated the effect of spatial vari - slide (Wang et al. 2022). Displacement at remolded shear ability of undrained peak and remolded shear strength on strength (δ ) also controls the post-failure behavior (Dey the retrogression distance and showed that spatial vari- et al. 2015; Islam et al. 2019). Dey et al. (2015) reported ability alters the distance of retrogression. He concluded that an increase of δ changes the failure pattern from that heterogeneity significantly affects the initiation and spread to flow slide. Increased δ results in an increase failure mechanism, and a deterministic analysis may in remolding energy (area under the stress–strain curve result in erroneous outcomes. up to remolded shear strength and strain) as well as a decrease in softening modulus. Simulation results show that increased remolding energy decreases the retrogres- Eec ff t of field conditions on the occurrence and post‑failure sion (Islam et al. 2018) and increased softening modulus behavior increases the retrogression (Locat et al. 2013; Wang et al. The impact of field conditions on landslides is not 2016a, b); that is, a higher δ will result in lower retrogres- straightforward and is often very complex. Landslides sion (Dey et al. 2016a). will respond to the combined effect of several hydrologi - Wang et al. (2022) assessed the effect of stability num - cal and hydrogeological parameters (Cloutier et al. 2016). ber (N ) on the occurrence of retrogressive failure and Therefore, establishing a direct correlation between suggested that retrogressive failure doesn’t occur for specific parameters and landslide occurrence is chal - N = 3.8 but occurs for N = 4.9 and 6.8. They concluded lenging and difficult to incorporate into the numeri - s s that sites characterized by a low stability number are cal analysis. Hence, studies on this issue are scarce and less prone to large retrogression than sites with a high highlight the need for more research. Wang et al. 2021) stability number. Earth pressure at rest (K ) also affects simulated the steady-state seepage condition with his oi the extent of retrogression. Slopes with higher K are strain rate-dependent strain softening constitutive soil oi susceptible to large progressive spreads rather than flow model (Eq. 4). Pore water pressure and seepage forces are slides (Locat et al. 2013; Wang et al. 2022). Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 21 of 28 calculated using a thermal–hydraulic analogy to model δ . A physical method to determine this parameter the in-situ stress condition for retrogressive failure in on the field or laboratory scale is required. sensitive clay slopes. They concluded that seepage signifi - • Constitutive models used in the literature to account cantly influences landslide triggers. An elevated artesian for sensitive clay behavior are formulated in a total pore pressure close to the slope’s toe may initiate sub- stress framework which is more relevant to account- stantial failure. A rise in the earth pressure coefficient, ing for undrained shear strength behavior. The devel - coupled with an increase in shear strength, can alter the opment of advanced effective-stress-based consti - failure pattern from flow slide to spread and thus affect tutive models is essential to evaluate in a unified the retrogression distance. coupled hydro-mechanical numerical framework long-term triggering factors with short-term rapid post-failure runouts without the need to predefine Final discussion the change in calculation mode from drained to und- In this study, the distinct characteristics of sensitive clay rained analysis. landslides and the advancement for modeling those land- • Most of the research has been focused on landslides slides have been discussed in detail, focusing on the land- initiated by toe erosion, whereas research simula- slide types, constitutive models, numerical frameworks, tion related to seismic loading and other triggering and critical factors controlling landslides mechanism. It factors is scanty. The influence of different field and is observed that, the unique mechanism of sensitive clay weather conditions on failure initiation and post- landslides, i.e., its progressive or retrogressive nature is failure behavior is still largely unknown and warrants completely attributed to the strain softening nature of further studies. the soil. Whether the failure advances as successive rota- • All the numerical studies of sensitive clay landslides tional flowslide or a translational movement resulting in have been done in plane-strain conditions. Still, in spread or a combination of both, it is always a result of reality, spatial variability in soil parameters could the movement of the liquified clay. Therefore, the most affect the failure mechanism to a great extent. Fur - essential component for the modeling of sensitive clay ther investigation of 3D effects is required. slopes is the constitutive soil model, which reproduces • There is scope for implementing advanced con - realistic strain-softening characteristics for accurately stitutive models and geometries in large deforma- capturing the complex landslide mechanism in sensi- tion numerical frameworks. These tools need to tive clays. Though several numerical frameworks are be further exploited in sensitive clay landslides. For well-suited to capture the large deformation associated instance, constitutive models with thermo-hydro- with retrogressive failure, the constitutive models need mechanical formulations can assess the thermal much improvement to represent the accurate post-failure effects on the creep behavior of sensitive clays (Li behavior of sensitive clays. The different numerical mod - et al. 2018) and the mobility of retrogressive rock els concerning sensitive clay landslides have been com- failures (Pinyol et al. 2018). It is essential to evalu- piled in Table 1. The table presents a list of literatures on ate whether these models also apply to sensitive clay the numerical modeling of sensitive clay landslides with landslides. the constitutive model and numerical framework used, • A limited amount of work has been performed the type of landslide that has been modeled, and the to evaluate the effect of geometrical and material outcomes of each study to give a general idea about the parameters for the prediction of different types of recent advancement and challenges in simulating land- retrogressive/progressive failures in sensitive clays. slides in sensitive clays. Further understanding of the critical parameters is Based on the review of the existing works on the essential for landslide prediction and risk assessment numerical modeling of sensitive clay landslides, some of such events. areas that require further study are noted below. Conclusion • The most advanced constitutive model discussed in This study presents an elaborate review of failure Sect. 4.1 is by Wang et al. (2022), which considered mechanisms, constitutive soil models, and numerical the strain rate effects, including the rheological prop - frameworks for the simulation of sensitive clay land- erty of soil, non-linear strength degradation, and var- slides. Additionally, the results from numerical stud- iation of effective stress with depth. The accuracy of ies have been compiled in a summary table. It has been this model vastly depends on the large-deformation found that modeling landslides with sensitive clays is parameter, displacement at 95% strength degradation Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 22 of 28 Table 1 Summary of recent studies for numerical modeling of sensitive clay landslides Name of the researcher Constitutive model with Numerical framework Type of sensitive clay Triggering factor Contribution on Limitations strain softening for large deformation landslide prediction of landslides 1. Locat et al. (2013) NGI-ANISOFT constitu- BIFURC Spread Erosion • Steeper slopes with • Linear strain-softening tive soil model with linear high K , slope with low • Strain rate effect and oi strain-softening stiffness, low remolded depth-wise variation of shear strength, high rate shear strength not con- of strain-softening are sidered more susceptible to large • Rheological property of progressive failure remolded clay not consid- • Spread occurs in highly ered over-consolidated clays • Homogeneous slope • No demonstration for dislocation of soil mass or formation of horst and grabens 2. Dey et al. (2015) Von-mises with exponen- Coupled Eulerian–Lagran- Spread Erosion • Illustration for the forma- • Strain rate effect and tial strain-softening gian tion of horst and grabens depth-wise variation of • Steeper slope, higher shear strength not con- sensitivity, lower δ are sidered ld more likely to cause large • Rheological property of progressive failure remolded clay not consid- • Increase in δ , S, H , ered ld t St and decrease in s of the crust changes the failure pattern from spread to flow slide 3. Dey et al. (2016a, b) Combined Erosion + surcharge load • Instantaneous velocity and surcharge load accel- erate the propagation of the shear band, causing progressive failure 4. Wang et al. (2016b, Von-mises with linear Implicit & Random mate- Retrogressive Gravity load • Failure surface forms in • Linear strain-softening 2016a) strain-softening rial point method the weakest soil layer • Strain rate effect and • Shear band propagation depth-wise variation of is governed by residual shear strength not con- strength sidered • For S = 5 the failure is • Rheological property of translational forming horst remolded clay not consid- and grabens ered • Spatial variability alters • Homogenous slope landslide initiation and • No clarification on the fail- propagation ure mechanism concerning failure type Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 23 of 28 ff Table 1 (continued) Name of the researcher Constitutive model with Numerical framework Type of sensitive clay Triggering factor Contribution on Limitations strain softening for large deformation landslide prediction of landslides 5. Zhang et al. (2018), Bingham-Tresca with linear Particle finite element Flow slide Erosion • Illustration for the multi- • Linear strain-softening Zhang et al. (2020, 2017) strain-softening method ple rotational slides • Depth-wise variation of • Ignoring the viscosity of shear strength not con- the remolded clay over- sidered estimates the retrogres- • Homogeneous slope sion and runout • Minimum sensitivity is required to initiate flow slide • Higher sensitivity increases the retrogression and runout • Eect of viscosity is higher for highly sensitive clays • Flow slides can result in horst and grabens 6. Tran and Solowski Tresca with exponential Material point method Spread Erosion • Sensitive clay slope • Rheological property of (2019) strain-softening with strain with S > 25 and τ <2 kPa remolded clay not consid- t ld rate effect are susceptible to large ered progressive failure • Strain rate has a sig- nificant impact on the propagation of progres- sive failure 7. Islam et al. (2019) Tresca with exponential Coupled Eulerian–Lagran- Flow slide Seismic loading • Steeper and inclined • Strain rate effect and strain-softening gian slopes have increased depth-wise variation of retrogression and runout shear strength not con- • Upslope surcharge load sidered increases retrogression • Rheological property of and runout remolded clay not consid- • Increased thickness ered of highly sensitive clay • Assumption of static changes the failure pat- stress–strain behavior of soil tern from spread to flow under dynamic loading slide • Increased remolding energy decreases retro- gression • A combination of rotational flow slide and translational spread can be possible in a large retrogressive failure Urmi et al. Geoenvironmental Disasters (2023) 10:14 Page 24 of 28 Table 1 (continued) Name of the researcher Constitutive model with Numerical framework Type of sensitive clay Triggering factor Contribution on Limitations strain softening for large deformation landslide prediction of landslides 8. Wang et al. 2021, 2022, Tresca with exponential Coupled Eulerian–Lagran- Flow slide and Spread Erosion • The occurrence of flow • No conclusive relationship strain-softening and rate gian slide or spread depends between remolding energy effect on the movement of liqui- and retrogression fied debris, brittleness of • Increasing stability soil, lateral earth pressure number did not result in • Lower strain rate increasing retrogression increases the mobility of • Some estimated input the debris leading to a parameters (δ , β, η) for large flow slide the constitutive model had high uncertainty 9. Zhang et al. (2018), Strain-softening Tresca Smoothed particle finite Retrogressive failure Erosion • Suitability of SPFM for • Linear strain-softening Yuan (2020) Model element method (SPFEM) retrogressive failure • Strain rate effect on shear • Increased softening strength not considered modulus increases retro- • Rheological property of gression and runout remolded clay not consid- ered • Homogenous slope 10. Shan et al. 2021 Elastoviscoplastic model Re-meshing and inter- Retrogressive failure lead- Decreasing shear strength • Increased S , decreased • Linear strain-softening polation technique with ing to spread of soil τ , decreased viscosity • Homogenous slope ld small strain (RITSS) of the remolded clay and increased riverbed width increases the retrogression Ur mi et al. Geoenvironmental Disasters (2023) 10:14 Page 25 of 28 Received: 25 December 2022 Accepted: 8 May 2023 extremely challenging, mainly due to the distinctive soil behavior. 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Geoenvironmental Disasters – Springer Journals
Published: May 26, 2023
Keywords: Progressive landslide; Sensitive clay; Numerical modeling; Strain-softening; Constitutive soil model; Large deformation
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