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GEOLOGY, ECOLOGY, AND LANDSCAPES 2020, VOL. 4, NO. 2, 151–158 INWASCON https://doi.org/10.1080/24749508.2019.1608408 RESEARCH ARTICLE Numerical modeling of translational dynamics for shallow landslides based on ﬂume tests – special case of spherical-cap-shaped slope sections a b J. Kanule and W. Ng’etich a b Department of Physics, University of Eldoret, Eldoret, Kenya; Department of Soil Science, University of Eldoret, Eldoret, Kenya ABSTRACT ARTICLE HISTORY Received 12 September 2018 Slope failures can endanger human life and cause infrastructural destruction and socioeco- Accepted 13 April 2019 nomic loss. Geoscientists have strived to develop constitutive models and real-time slope monitoring models and systems to abate these processes. Most research studies have KEYWORDS proposed models which describe the dynamics of wedge-shaped soil masses which do not Translational failure; mimic real ﬁeld conditions. In this study, failure dynamics of spherical-cap-shaped soil masses numerical model; shallow on an inclined slope section undergoing purely translational displacement are described landslides using empirical models derived from inertial forces in action for varying hydrological condi- tions. Validation of model results was done through experimental tests carried out on a laboratory ﬂume. Empirical models representing rainfall intensity, soil water content, pore- water pressure, factor of safety, and displacement were derived. More pertinently, the empirical model for the factor of safety is derived considering the moist unit weight of the soil as opposed to earlier models which focused on saturated conditions only. Model and experimental results indicate close concurrence, especially for the factor of safety with root mean square error of 0.0385 and r of 0.6381. Since the models are physics based, they can be applied on a variety of rainfall-induced shallow landslides on relatively steep slopes. 1 Introduction 2003). Fang, Cui, Pei, and Zhou (2012) reported that shallow landslides occur as a result of sudden Many incidences of soil mass movements including loss of shear resistance when high pore-water pres- but not limited to landslides or mudslides have been sures develop leading to liquefaction due to cyclic reported lately in many parts of Kenya (especially in loading. While high-intensity, short-duration rainfall the Rift valley, central and western highlands), claim- events are known to trigger soil mass movements, ing a sizeable number of populace and enormous even long-duration, low-intensity rainfall and rapid destruction to infrastructure leading to general socio- snow or ice melt also activate landslides (Guzzetti, economic meltdown. Sloping regions are considered Ardizzone, Cardinali, Rossi, & Valigi, 2009). as environmentally sensitive areas where many land Earthquakes of magnitude greater than M = 4.0 may development activities for agricultural production, generate strong ground shaking movements which agro-tourism, property development, and road con- rapidly reduce the frictional strength and/or increase struction projects are ongoing with high prospects of shear stress of hillslope material through rock mass other infrastructural expansions. While the impacts shattering or liquefaction, thereby triggering land- of soil mass movement incidences have had negative slides (Meunier, Hovius, & Haines, 2008). insinuations on the socioeconomic proﬁle of the Several slope dynamic models including Janbu, regional general development, there exist a number Mertens, and Bishops have been utilized in many of gaps in terms of the theoretical formulation, mon- cases to provide an analytical equation relating cer- itoring, and mitigative measures. tain parameters of the physical system to its tem- For any form of soil mass wasting to occur, certain poral behavior in space. All these models highlight precursory factors set the stage for triggering events the factor of safety as an indicator of the health that initiate failure (Rybar, Stemberk, & Wagner, status of theslope foragivensetof conditions 2002). The most common triggers of landslides are (Najjar, Ali, & Basheer, 1999). Incidentally, there precipitation, seismicity, and human activities (Petley, exist some cases where these models are inapplicable 2009). According to earlier studies, water reduces because of the inherent nonlinearity of the system, shear strength of a soil mass either by accelerating lack of experimental information, experimental the process of crack formation and development or inaccuracy, and deviations from the ideal conditions lubricating the soil grains by ﬁlling the pore spaces (Shahin, Jaksa, & Maier, 2001). In addition, since (Anderson & Anderson, 2010; Kilburn & Petley, many factors are used as inputs in modeling slope CONTACT J. Kanule firstname.lastname@example.org Department of Physics, University of Eldoret, P.O. Box 1125-30100, Eldoret, Kenya © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 152 J. KANULE AND W. NG’ETICH stability, most of these models cannot replicate phy- of the volume of a spherical cap, we derive the weight sical occurrences in nature, especially when describ- of a dry soil skeleton of nearly the same shape as ing vital factors such as slope geometry and soil πγ H cos α 3L properties aﬀecting the stability of slopes (Jia, d 2 2 W ¼ þ H cos α (1) Zhan, Chen, & Fredlund, 2009). 6 4 Soil mass movement processes are studied with where γ is the dry unit weight of the soil deﬁned by a view to develop physical models based on geomor- phological factors (slope gradient, aspect, and relative Gγ relief), soil characteristics (depth, structure, perme- γ ¼ (2) 1 þ e ability, and porosity), and hydrological factors, which can be employed in the characterization of a wide with G, the speciﬁc gravity of soil; and e, the void ratio. range of slope sections. In this study, empirical phy- For a soil mass under wetting conditions usually sics-based models are derived from ﬁrst principles through a rainfall simulator or irrigation event (intro- based on inertial forces that build up when a soil duction of water into the soil crystal matrix), the new mass is inclined at an angle under modest wetting weight of the saturated soil segment assuming there is conditions. The objective of the study is to investigate negligible run-oﬀ is translational motion beginning from inertial forces acting on a soil mass, to the buildup of pore- πγ H cos α 3L e 2 2 W ¼ þ H cos α (3) pressures and shear stresses to the eﬀect of these 6 4 forces on the factor of safety and by extension the displacement downslope. where γ is the eﬀective unit weight of the soil deﬁned by De Vleeschauwer and De Smedt (2002)as q cos α 2 Materials and methods γ ¼ þðÞ 1 m γ þ mγ (4) e d s 2.1 Model formulation for which q is the additional weight on the soil sur- A numerical model is proposed that describes the face by vegetation or structures. The factor m in dynamics of a soil mass inclined at an angle, α.In Equation (5) is the wetness index deﬁned by Ray, this study, we consider the simpliﬁed case of a nearly Jacobs, and de Alba (2010)as concave-shaped slope section that resembles a spherical cap of relatively small height inclined at θ h þðÞ H h h þðÞ H h S m ¼ ¼ (5) an angle of α to the horizontal and of approximately H H inﬁnite lateral extent as depicted in Figure 1. The soil mass is considered as a homogeneous rigid-perfectly where h is the saturated thickness of the soil above plastic material which undergoes shear failure when the failure plane and S= θ/n is the degree of satura- driving and frictional forces are not balanced. The tion; n is the soil porosity and θ is the volumetric volume in consideration is illustrated by the portion moisture content derived from the modiﬁed soil– abcd which is approximated to a spherical cap of water characteristic curve proposed by Fredlund and height Hcosα and base length L. From the deﬁnition Xing (1994) as Figure 1. Schematic representation of a spherical-cap-shaped slope section (shaded) and the inertial forces acting on it. GEOLOGY, ECOLOGY, AND LANDSCAPES 153 2 3 2 3 while the soil cohesion which is a function of the ln 1 þ ψ θ r 6 s 7 water content and soil lithology has been empirically 4 5 θ ¼ 1 (6) 4hi hi 5 10 ψ modeled as ln 1 þ ln e þ 0 1 c ¼ aθ e (14) where θ is the saturated volumetric water content, ψ is the soil suction, ψ is the residual soil suction, e is r n where a is a curve ﬁtting parameter, while b is a factor a natural number, while β,p, and q are curve ﬁtting related to the observable soil suction for water. parameters with β carrying the units of pressure. Assuming that there is minimal seepage through For the model proposed by De Vleeschauwer and the soil mass in consideration and that the ground- De Smedt (2002) (Equation 4), the eﬀective unit water level in this segment is parallel to the incline weight of the soil is valid with its assumption that plane, i.e., coincides with the ground surface adjacent water pressures in the wet pores are transmitted to to it, then the shear stress of the soil with eﬀective / / the failure plane through the interconnected wet soil cohesion c and eﬀective angle of shear resistance ϕ is pores. In contrast, we propose that for more precise given by (Bishop, 1967): results, the unsaturated zone soil moisture content 0 0 τ ¼ c þðÞ σ u tan ϕ must be accounted for in the computation of the wetness index and the eﬀective unit weight as opposed or to earlier studies where the eﬀective unit weight of the τ ¼ c soil was considered either for purely saturated condi- 2γ Hcos α 3 H tions or on purely dry soil skeleton. Based on earlier em 2 2 0 þ þ cos α γ Hcos α tan ϕ studies by Sidle and Ochiai (2006), we propose sub- 3 4 L stitution of the dry unit weight with the moist unit (15) weight (γ ) in Equation (4) resulting in Assuming that slope stability is characterized by the q cos α Mohr–Coulomb failure criterion and that there are γ ¼ þðÞ 1 m γ þ mγ (7) em m s no external loads, the factor of safety will then be where computed by γ ¼ GγðÞ 1 nðÞ 1 þ θ (8) 2c csc 2α 1 tan ϕ m w w FS ¼ þ 1 (16) Θγ H Θ γ tan α em em is the moist unit weight. 2 3 h 2 The total weight of moist unsaturated soil (W ) where Θ ¼ þ cos α . moist 2 3 4 L will now be obtained by utilizing Equation (7) as The acceleration of the soil mass downslope has been modeled by the relation πγ H cos α 3L em 2 2 W ¼ þ H cos α (9) moist a ¼ gΦð1 FSÞ½ sin α cos α tan ϕ (17) 6 4 where Φ is a curve ﬁtting parameter. This, therefore, Consequently, by deﬁnition, we derive the eﬀective leads to the derivation of the equations of motion for normal and shear stresses for moist soil mass, respec- velocity and displacement components, respectively, as tively (considering the base area = πL /4) as 2 2 2γ Hcos α 3 H v ¼fg 2gHΦð1 FSÞðÞ sin α cos α tan ϕ (18) em 2 σ ¼ þ cos α (10) 3 4 L S ¼ (19) and 2a 2γ H cos α sin α 3 H em 2 τ ¼ þ cos α (11) 3 4 L 2.2 Experimental setup The proposed empirical model for precipitation for The experimental tests were conducted in the Soil a given period of time is derived from an odd Fourier Science Lab at the University of Eldoret. Soil samples Series expression as used in the experiments were taken from Sergoit swamp k sinðÞ 2η 1 t with predominantly ferralic cambisol soils. A solar- RnðtÞ¼ (12) η ðÞ 2η 1 powered monitoring (SPM) system was fabricated com- η¼1 prising a model ﬂume installed with both electronic and where k and η are curve ﬁtting parameters. optical displacement sensors connected to a data acqui- The pore-water pressure in the soil mass for sition panel. The dimensions of the model ﬂume were a piezometer placed at point h is given by 1.6 m long, 0.6 m width, and 0.5 m high (Figure 2). The model ﬂume was made up of a metallic framework with u ¼ γ h ¼ γ Hcos α (13) w w wooden sheets on the sides. 154 J. KANULE AND W. NG’ETICH Control panel and transmitter Rain gauge VW Piezzometer Soil moisture sensor Ultrasonic sensor Figure 2. SPM system setup. A rainfall simulator consisting of a water source, performance of the models against laboratory results nozzle array, and a ﬂow rate controller was used. It (Nahm, 2016). The RMSE is the measure of the diﬀer- was mounted on the system in such a way that the ence between values predicted by a model and the ﬂow rate could be controlled remotely via an electronic actual experimental values deﬁned mathematically as sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ switch. Monitoring of the incident rainfall was realized j 2 ðX X Þ via a Steven’s rain gauge which was wirelessly connected obs;i model;i i¼1 RMSE ¼ (20) to a Steven’s Vantage Console for onward transmission j of data to a remote computer. Pore-water pressure in where X and X are observed and modeled obs model the soil mass was measured by vibrating wire piezo- values at time/place i, respectively, while j is the meter connected to a 4–20-mA data logger. Soil moist- number of data points. ure content and displacement were monitored using On the other hand, the correlation coeﬃcient indi- Arduino-based resistivity and ultrasonic transducers, cates the strength and direction of a linear relation- respectively. Arduino-based sensors were connected to ship between model output and experimental values. a microprocessor for interface with a PC. A ﬂowchart of For a series with i observations and j model values, data from the array of transducers to the data loggers the correlation coeﬃcient is used to estimate the connected to the remote server is shown in ﬁgure 4. correlation between model and observations as Two experiments were carried out with a rainfall simulator, while one control system was set up with no ðx x Þðx x Þ obs;i obs model;i model i¼1 r ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rainfall applied to it. For each soil sample collected, P P j 2 j 2 ðx x Þ ðx x Þ obs;i obs model;i model approximately 80% of it (≈120 kg) was placed in the i¼1 i¼1 model ﬂume to form a single ﬂat layer and then the (21) remaining 30 kg was poured at the center to form the If the correlation is +1, it indicates the case of a perfect spherical-cap-shaped mass. No compaction of the soil increasing linear relationship and −1 if it is a decreasing was conducted. One side of the ﬂume was then tilted linear relationship. Values in between +1 and −1indi- at an angle of 58° to the horizontal using a high-lift cate the degree of linear relationship between two sets of jack. Failure of the soil mass was facilitated by the use observations. A correlation coeﬃcient of 0 implies there of an artiﬁcial rainfall from the simulator. The sphe- is no linear relationship between the variables. The rical-cap-shaped model of the slope was chosen from square of the correlation coeﬃcient (r ) indicates how the fact that, for a given slope of relatively high gra- much of the variance between the two variables is dient with negligible vegetation cover, due to uneven described by the linear ﬁt. erosion incidences, moderate bump-shaped soil masses are left behind which resemble spherical caps of ﬁnite length along the incline plane. 4 Results and discussion The use of computational methods in determining soil mass dynamics makes it possible to evaluate, with 2.3 Data analysis asuﬃcient degree of precision and in a simple man- Statistical error estimates such as root mean square ner, the characteristics of a given slope under condi- error (RMSE), coeﬃcient of correlation, and coeﬃcient tions of intense but periodic rainfall or irrigation of determination (r ) were employed to analyze the events modeled from simple physics-based equations. GEOLOGY, ECOLOGY, AND LANDSCAPES 155 Data from numerical results arising from the derived As soil moisture content rises as a result of inﬁl- model are then compared with experimental data tration, the negative pore-pressures will remain obtained from the model ﬂume for purposes of valida- almost constant in the initial stages because not all tion. Although the validation samples were indepen- pore spaces are ﬁlled. But after some time (for our dent of in-situ ﬁeld conditions because they were case 30 min), there is an exponential rise in the pore- disturbed in one way or the other during sample pressure towards the positive phase as saturation collection, the geology and soil properties are kept conditions approach (Figure 3(c)). Computational unchanged. Three samples were tested, one of which results from the model agree well with the experi- was a control experiment. The results of each para- mental data except that after failure, the model envi- meter measured were recorded and an average com- sages a faster drop in pore-pressures, but in the puted that was then used in the model for comparison. experiment, they remain almost constant for some Using a rainfall simulator, incident rainfall event time since the soil mass must undergo a drying pro- was maintained at 45 mm for 45 min and then cess. It is also observed that the pore-water pressures stopped for all the experiments. This condition was in the soil mass continue to rise even when the rain- adhered to in order to study the eﬀects of pore-water fall event has been halted, indicating their strong pressures on the soil mass and its dependence on the dependence on inﬁltration rate rather than rainfall soil moisture content. Computational results from the intensity directly. This means that pore-pressures proposed model (Equation 11) for a single rainfall will, therefore, vary according to the moisture content event as a function of time were compared to experi- present in the soil at any given time regardless of the mental data as illustrated in Figure 3(a). From the source of water whether irrigation event or rainfall. plot, the RMSE and Pearson correlation coeﬃcient of Numerical model results as compared to experimen- 3.9385 and 0.9819, respectively, were obtained tal data produced RMSE and correlation coeﬃcient of between model results and experimental laboratory 0.5473 and 0.9261, respectively, which indicates data. This implies that the accuracy of the proposed a close concurrence. model is relatively high. Addition of water into a soil mass on a slope In this experiment, it is assumed that inﬁltrating through rainfall inﬁltration or irrigation event serves water serves to increase the weight of the soil mass to increase the weight of the sliding plane and lubri- (and by extension the normal stress) and alter the cating the soil particles, thereby increasing the driv- cohesion depending on the degree of saturation. The ing forces downslope and/or signiﬁcantly reducing variation of the moisture content over time for both the shear resistive forces consequently leading to model and experimental data is illustrated in Figure 3 a drop in the factor of safety. The value of the factor (b). The amount of moisture content in the soil mass of safety indicates the health status of a given slope. is a function of the speciﬁc moisture capacity, speciﬁc Values greater than unity indicate higher shear storage (computed as the inverse of the soil skeleton strength, while values lower than one point to bulk modulus), pore-water pressure (negative in a very unstable slope. The factor of safety is unsaturated zone), time elapsed, relative permeability, a function of cohesion, moisture content, pore- dynamic viscosity of water, and vertical elevation pressure, internal friction angle, and slope angle. coordinate (van Genuchten, 1980). Volumetric soil Figure 3(d) shows a comparison of both numerical moisture content is found to rise from a modest and experimental trend of a slope under gradual value of 20% to nearly saturation, i.e., about 93%, wetting conditions. In the ﬁgure, an increase in when the soil mass begins to move downslope. This moisture content which fuels an increase in negative is observed from the development of numerous pore-pressures together with a notable drop in cohe- cracks which begin to coalesce into bigger ones as sion will serve to lower the factor of safety to below moisture content increases steadily. Because of the unity, a state that exacerbates the slope to imminent high plastic strength of the type of soil sampled, failure. The factor of safety is observed to drop below a very high degree of saturation was required to unity when the pore-pressure rises to the positive reach failure. Additionally, during the experiment, phase. As alluded to earlier, pore-pressures may still the rainfall simulator was programmed in such rise even after a rainfall event, as in our experiment. a way that when pore-water pressures began to rise, Correspondingly, the factor of safety may also drop to it could be stopped automatically. While the model below unity even after the rainfall event. In this case, assumes water content steadily drops after the rainfall the factor of safety dropped to below unity after event, experimental results deviate slightly showing approximately 50 min of the experiment. This a nearly exponential decrease, and this is attributed to explains why most slopes collapse some time after the speciﬁc storage factor for the speciﬁc soil mass. a rainfall event and not during the storm. Model Performance comparison of the numerical model to and experimental results are in agreement except for experimental data yielded an RMSE and r of 0.0108 a negligible number of experimental deviations with and 0.9291, respectively. RMSE of 0.0385 and r of 0.6381. 156 J. KANULE AND W. NG’ETICH 40 y = 1.0159x - 0.8833 R² = 0.9641 Rn (mm)-control 025 50 20 Rn (mm)-Model Rainfall - Experiment Rn (mm)-Expt Water Content (Control) Water Content (Model) Water Content (Expt) y = 1.0389x - 1.5476 R² = 0.9291 20 60 100 Water Content (Experiment) y = 0.9906x + 2.6601 R² = 0.9261 -40 -20 0 20 -20 -40 Pu-control Pu (Experiment) Pu (Mpa)- model -15 Pu (Mpa)-Expt -30 -45 1.6 1.2 y = 1.1654x - 0.1241 R² = 0.6341 1.4 0.8 1.1 FS (Control) 0.4 FS (M odel) 0.8 0.8 1.1 1.4 FS (Expt) FS (Experiment) disp (Control) disp (M odel) disp (Expt) y = 1.0577x + 0.9928 R² = 0.6818 010 20 Disp.(Expt) 030 60 90 Time (minutes) Figure 3. Computational and experimental results of (a) rainfall intensity; (b) water content; (c) pore-pressure; (d) factor of safety, and (e) displacement, as a function of time in minutes. Inset: statistical analysis. Pore-water Pressure (Mpa) Rainfall (mm) Volumetric Water Content (%) Displacement (cm) Factor of Safety Disp.(Model) FS (Model) Pu (Model) Water Content (Model) Rainfall - Model GEOLOGY, ECOLOGY, AND LANDSCAPES 157 Rain gauge Davis Weather Barometer Station Hygrometer Vantage Pro2 Anemometer Console (Data Logger) SM Rad Air Digital Internet io Thermometer Pyranometer Wireless Soil Moisture Node probe Soil Temperature probe Data Processing Geophone Data 4-20 mA Acquisition Module Module VW Piezzometer Remote Server Sensor Ultrasonic Module Range Sensor Figure 4. SPM ﬂowchart. The variation of factor of safety over time dic- 5 Conclusion and recommendations tates the dynamics of a given slope. According to The proposed numerical models regarding translational the Mohr–Coulomb failure criterion, the probability failure for a spherical-cap-shaped slope section have of translational failure is very high for any slope been shown to agree well with the experimental results with values of factor of safety less than unity. Slope obtained from measurements using the SPM system, failure can be translational or circular. For relatively speciﬁcally for shallow soil masses at steep slopes. The steep slopes, translational displacement is more proposed model for the factor of safety and by extension probable as observed from these experiments. the other hydrological models derived are unique in Because of erosion occurrences, especially for bare that they take into consideration the moist soil unit land, the resulting slope shape usually appears like weight as opposed to earlier models which were applied a spherical cap with its diameter lying along the only in extreme conditions of purely dry soil or satu- incline plane. When inﬁltration proceeds in this rated conditions. The model is also more convenient as spherical-cap-shaped slope, part of the water it contains fewer variables as many of them are com- increases its weight, while the other part creates puted as empirical functions of water content. This a weak section along the incline plane through model of factor of safety is convenient for shallow land- lubrication for nearly saturated conditions. Under slides at relatively steep slopes. these conditions, the cohesive strength of the soil Since the study was conﬁned to one particular mass will be lost, and the downward gravitational angle throughout the experiment, we, therefore, forces will exceed frictional forces leading to down- recommend further investigations on the accuracy ward displacement. In the process, the soil mass and reliability of this model at diﬀerent angles. will either be displaced as a whole but eventually Similarly, diﬀerent soil types should be tested to break into smaller lumps or undergo liquefaction ascertain the reliability of these models. We also and ﬂow as a liquid down the plane, depending on recommend an experimental testing of this model in thetypeofbeing tested.Forourcase, thesoilmass the in-situ conditions at diﬀerent locations. was undergoing liquefaction for all cases tested. Figure 3(e) illustrates the displacement characteris- tics ofthesoilmassovertimewithacomparison of Acknowledgments the numerical model and experimental results. We acknowledge the support of Physics Department, Soil Again, model results marry well with the experi- Science Department, University of Eldoret, and National mental ﬁndings depicting the accuracy of the 2 Council for Science and Technology (NCST). model, i.e., RMSE as 0.1496 and r of 0.8257. 158 J. KANULE AND W. NG’ETICH Kilburn, C. R. J., & Petley, D. N. (2003). Forecasting giant, Disclosure statement catastrophic slope collapse: Lessons from Vajont, No potential conﬂict of interest was reported by the Northern Italy. Geomorphology, 54(1-2), 21-32. authors. Meunier, P., Hovius, N., & Haines, J. (2008). Topographic site eﬀects and the location of earthquake induced landslides. Elsevier: Earth and Planetary Science Letters, 275, 221–232. ORCID Nahm, F. S. (2016). 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Geology Ecology and Landscapes – Taylor & Francis
Published: Apr 2, 2020
Keywords: Translational failure; numerical model; shallow landslides
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