Run-out of landslides in brittle soils

Run-out of landslides in brittle soils 1 A. Yerro1 & E.E. Alonso1,2 & N.M. Pinyol1,2 2 1Department of Geotechnical Engineering and Geosciences, UPC, Barcelona, Spain 3 2Centre Internacional de Mètodes Numèrics en Enginyeria 4 5 ABSTRACT: One of the factors causing the acceleration of landslides is the loss of strength of 6 the soil involved in the potential unstable mechanism. The travelled distance and the landslide 7 velocity, a key factor in risk analysis, will be determined by the loss of resistant forces. Brittle 8 behaviour, commonly associated with cemented soils, overconsolidated plastic clay formations 9 and sensitive clays, lead to the progressive failure phenomenon explained by the reduction of 10 the strength with increasing strain. In the present study, this phenomenon has been analysed in 11 the case of a saturated slope which becomes unstable by increasing the boundary pore water 12 pressure. A Mohr-Coulomb model with strain softening behaviour induced by increasing 13 deviatoric plastic strain is used. The paper focusses not only on the stability of the slope but also 14 on the post failure behaviour (run-out and sliding velocity). A coupled hydro-mechanical 15 formulation of the Material Point Method has been used to simulate the whole instability 16 process. The influence of the brittleness of the material on the triggering of instability and run-17 out is evaluated by means of a parametric study varying peak and residual strength. The onset of 18 the failure and the failure geometry are controlled by both peak and residual values. Good 19 correlations between run-outs and brittleness are found. The decay of the strength determines 20 the acceleration of the landslides and the travelled distance. 21 1 INTRODUCTION 22 The dynamic behaviour of landslides receives increasing attention because landslide risk 23 analysis and spatial identification of vulnerable areas require estimations of the slide run-out 24 and the velocity of the unstable mass [1]. Special attention is given to reservoirs, lakes and 25 fjords potentially affected by landslides on their margins [2–4]. In fact, slope instabilities may 26 affect dams and their foundations and they may lead to partial or complete blockage of rivers, 27 creating dangerous “natural” dams or the generation of a destructive wave due to the impact of 28 the landslide against the stored water [5–7]. The potential damage caused by landslides can be 29 determined by several factors related with the volume of the mobilized mass, the run-out, 30 velocity and acceleration. One of the factors that control the acceleration of the slide is the loss 31 of resistant forces associated with the drop of available soil strength. This phenomenon is 32 typically observed in first time failure developed in “intact” sites in materials exhibiting a brittle 33 behaviour. This is the case of hard soils and soft rocks, overconsolidated and cemented clayey 34 soils with special relevance in the case of high plasticity soils. These materials exhibit a 35 softening behaviour from a peak value, associated with a low value of shearing displacements, 36 to a low residual strength when bonds are destroyed and clay particles orient in the direction of 37 shearing. This reduction of strength leads to the propagation of the failure surface following a 38 process of progressive failure. 39 When a point exceeds the maximum available strength, a degradation process initiates due to 40 the strain softening associated with the constitutive response of the material. The unbalanced 41 stresses are transferred to the surrounding areas which in turn may overstress neighbouring 42 points in the process, leading eventually to residual strength conditions. This stress transfer 43 phenomenon develops during slip surface propagation. This mechanism was first recognized by 44 Terzaghi and Peck [8] and Taylor [9]. It was further discussed in the context of 45 overconsolidated clays and clay shales by Skempton [10], Bjerrum [11] and Bishop [12]. 46 Further contribution are made by Palmer and Rice [13], Stark and Eid [14], and Puzrin and 47 Germanovich [15]. 48 Several real cases involving progressive failure are collected and analyzed in the literature [16–49 20]. Troncone [21] presents a 2D numerical analysis of well documented Senise large landslides 50 in Southern Italy and a 3D extension in [22]. Other real cases of landslides involving 51 progressive failure mechanism in the Iberian Peninsula have been collected in [23]. 52 Contributions mentioned above mainly concentrate on the analysis of the generation and 53 evolution of the failure surface but the run-out stage, once instability occurs, is not explored. 54 Modelling large displacement involves the use of alternative calculation techniques to the 55 Lagranian approaches generally used in FEM. Soga et al. [24] reviews current numerical 56 methods capable of analysing the slide motion. In this work, the Material Point Method (MPM) 57 [25] is selected to analyse the stability of slopes and their post failure response in strain 58 softening materials. MPM is a numerical technique able to simulate large displacements by 59 means of combining two discretizations of the media: (a) a set of material points which move 60 through (b) a fixed computational grid. This dual description prevents mesh distortion problems 61 and contacts between different bodies are automatically solved. 62 A fully coupled hydro-mechanical material point code was developed for saturated soils within 63 the MPM Research Community framework [26–29]. A strain softening elastoplastic constitutive 64 law was has been implemented with the purpose of analysing progressive failure phenomena 65 that take place in materials exhibiting a reduction of the strength with increasing strain [30]. 66 This MPM formulation was recently applied in Alonso et al. [31] to model the Selborne failure 67 experiment [32]. Failure of the Selborne slope was triggered by forced water recharge. Field 68 instrumentation data indicated that the failure was a progressive mechanism in overconsolidated 69 brittle clays. The numerical MPM analysis presented in [31] provided consistent and accurate 70 results in the prediction of the shape and position of the failure surface, the development of 71 progressive failure and the slide motion after failure. 72 The aim of the paper is to explore the response of saturated slopes in brittle materials. It 73 focusses on exploring the material properties controlling the run-out distance and velocity of the 74 unstable mass in brittle materials. First, a synthetic slope is presented in which the shear stress 75 distribution and the progression of failure mechanism are discussed. Afterwards, by means a 76 parametric analysis, the brittle behaviour of the material soil (defined in terms of brittleness 77 index IB) is shown to be a key factor of the slope response. The results are discussed with the 78 aim of deriving practical conclusions. 79 2 BASIS OF MPM FORMULATION 80 The MPM [33] discretizes the continuum as a set of subdomains. In the standard approach, 81 presented by Sulsky et al. [25], the mass of each subdomain is considered to be concentrated in 82 a point, the material point (Fig. 1). Other properties such as velocities, strains and stresses, are 83 also carried by the material points. This information is projected on to a background mesh 84 where governing equation are solved. The support computational mesh covers the full domain 85 of the problem and remains fixed during calculation. Calculations on the mesh serve to update 86 the material point properties and location. Linear interpolation shape functions are used to 87 provide the relationship between material points and nodes at any point of the domain. This 88 approach allows MPM to combine the advantages of Eulerian and Lagrangian formulations. 89 90 Fig. 1. Scheme of the spatial discretization used in MPM formulation. 91 The MPM formulation for a mechanical problem was presented by Sulsky and Schryer [34]. 92 Different authors have extended the MPM to solve coupled hydro-mechanical problems under 93 saturated conditions [20,27,35]. More recently, Yerro et al. [28] extended MPM for unsaturated 94 soils. 95 The numerical approach considered in this work to simulate saturated soils is based on [27]. It 96 assumes that each material point represents a portion of the soil, moves attached to the solid 97 skeleton and carries information of solid and liquid phases. Solid and liquid accelerations are 98 calculated in the computational mesh solving the dynamic momentum balances of both phases. 99 Velocities, displacements and strains are obtained in the material points; and liquid mass 100 balance equation is established in the material points to provide liquid pressures. An explicit 101 Euler-Cromer scheme [36] is used to update displacements and velocities from calculated 102 accelerations. 103 In order to avoid non-physical vibrations, it is common to include a damping term in the balance 104 equations. The approach adopted here was presented by Cundall [37]. It introduces a damping 105 force proportional to the corresponding out-of-balance force (proportional factor 𝛼) and 106 opposite to the phase velocity. In dynamic problems, the proportional factor should be very 107 small (0-5%) in order to approximate the correct solution and avoid an overdamped system. 108 The standard MPM approach [25,33] in which the mass of each material point is assumed to be 109 concentrated at the corresponding material point, suffers from spurious oscillations when 110 material points cross from one element to another one. It is caused by a jump discontinuity in 111 the gradient of low-order shape functions that are used for the integration. In order to reduce this 112 numerical problem, a simple technique of low computational cost is introduced in this work 113 [38]. It arises from considering that the stress on each element is constant and corresponds to the 114 average of the stresses of the material points located within a given cell. Other authors proposed 115 more accurate techniques. For instance, Bardenhagen and Kober [39] proposed to distribute the 116 mass of each material point in a certain region. This idea results in a family of methods known 117 as Generalized Interpolation Material Point (GIMP) methods. More recently, MPM has been 118 extended to convected particle domain interpolation methods (CPDI1 and CPDI2) which are 119 developed to improve the tracking of material point domains [40,41]. 120 3 CONSTITUTIVE MODELLING 121 In this paper the basic non-associated Mohr-Coulomb law is generalized to introduce strain 122 softening plasticity with the aim of modelling a strength loss after peak strength conditions. In 123 order to reduce the singularities of Mohr-Coulomb yield surface (edges and tip) that involve 124 some numerical problems during the elasto-plastic integration, the modifications proposed by 125 Abbo and Sloan [42] have been implemented. 126 Following previous contributions [43–47], the softening behaviour is accounted for by reducing 127 the strength parameters (friction angle φ’, and cohesion c’) exponentially with the accumulated 128 deviatoric plastic strain p d ε according to the following softening rules: 129 ( ) p d r pr c c c ce ηε − ′′ ′′ =+− (2) 130 ( ) p d r pr e ηε φφ φφ − ′′ ′′ =+− (3) 131 132 The deviatoric plastic strain invariant is defined as: 133 2 3 p pp d ij ij ε =ee (4) 134 where p ij e is the deviatoric part of the plastic strain tensor. 135 The model requires the specification of peak (cp’,φp’) and residual (cr’,φr’) effective strength 136 parameters. An additional parameter η, a shape factor parameter, is also necessary in order to 137 control the rate of strength decrease. 138 The effect of η in a simple shear test simulation is shown in Fig. 2. The soil parameters of the 139 material considered in these simulations are summarised in Table 1. A vertical stress of 50 kPa 140 and a horizontal one of 25 kPa are applied to confine the sample. Then, a prescribed velocity is 141 imposed at the upper boundary maintaining the bottom fixed. High values of η lead to faster 142 degradation of the soil strength. 143 144 145 146 Fig. 2. Evolution of shear stress in a numerical model of a simple shear tests for different values of 147 parameter η. 148 4 A REFERENCE SLOPE INSTABILITY PROBLEM 149 The instability of a synthetic slope, 6 m high and 37º steep, was analysed (Fig. 3). The slope 150 failure was triggered by increasing the pore water pressure at the lower boundary simulating a 151 phreatic level rise. This is a plane strain simulation in which the boundary conditions on the 152 vertical contours are rollers and the base is fixed. The water pressure is zero along the slope 153 surface, the lateral contours are impermeable and saturated conditions are considered during the 154 calculation. The mesh was refined in the region where the failure is expected in order to get 155 more accurate results and to optimise the computational cost. 156 Initially the slope remains in equilibrium. The calculation starts with the application of a 40 kPa 157 increase in pore pressure (ΔP) along the lower boundary during 1 second. Afterwards the water 158 pressure on the boundary is maintained constant during the entire simulation. 159 The Mohr-Coulomb strain softening model described in the previous section was used to 160 simulate the brittle behaviour of a soil. The properties of the slope material are given in Table 1. 161 The particular values selected are not relevant for the discussion presented here. The only 162 requirement to select such values has been to ensure that the failure occurs for the imposed 163 increment of pore water pressure. The effect of the shape factor parameter on the drop of the 164 strength is shown in Fig. 2. 165 166 Fig. 3. Scheme of the MPM model. Initial distribution of the material points and computational mesh. 167 Table 1. Soil parameters of the slope. 168 Soil parameters Value Porosity [-] 0.2 Intrinsic permeability [m2] 10-10 Young’s modulus [kPa] 20000 Poisson’s ratio [-] 0.33 Peak cohesion [kPa] 5 Residual cohesion [kPa] 0.5 Peak friction angle [º] 35 Residual friction angle [º] 20 Shape factor parameter 500 Dilatancy angle [º] 0 169 An explicit Euler-Cromer scheme is used to discretise the governing equations. Because it is 170 conditionally stable, very small time steps are required in the calculation. Since permeability is 171 not a relevant parameter in the analysis presented here, a high value (0.001 m/s) has been 172 adopted to simulate the slope failure in a relatively short time. 173 In order to reduce numerical instabilities a damping force has been included in the momentum 174 balance equation. It is proportional to the corresponding unbalanced force by means a 175 proportional factor α = 0.05. 176 The increase of pore pressure reduces the effective stresses in the slope leading some points to 177 reaching peak conditions. The strain softening effect decreases progressively the strength 178 parameters of the plastic zones down to the residual yield surface. As a result, the gravitational 179 stresses are sufficient to induce a progressive failure in the case analysed. 180 Failure development is illustrated in Fig. 4 by representing the shear strain contours at two 181 different times. At 8.3 s a shear band localises providing a failure mechanism and afterwards the 182 instability initiates. During the movement, the shear band spreads. Finally, when the new 183 geometry becomes stable, a wider shear zone is observed (Fig. 4b). Fig. 5 shows the final 184 displacements field. In this case, the achieved maximum displacement is 9.1 m and the 185 displacement between the toe of the initial and the final slope geometries is 9.5 m. 186 187 (a) 188 189 (b) 190 Fig. 4. Distribution of the shear strain at different times. Note the different scales of the shear strain. 191 192 Fig. 5. Distribution of the calculated total displacement at 25 s after the initiation of the failure. The 193 maximum displacement of the toe is also indicated. 194 Following Skempton [10], a mobilised friction angle (MFA) ˆ' ϕ is defined as 195 ( ) ϕ ϕ ′′ + = ′tan ˆ sin cp q (5) 196 Where q and p’ are the deviatoric and effective volumetric stress components defined as 197 follows: 198 2 31 σσ ′ − ′ =q ; 2 31 σσ ′ + ′ = ′ p (6) 199 This measure of the mobilised strength is used to analyse the stress evolution of points 200 homogeneously distributed along the initial failure surface. Note that ˆ' ϕ coincides with the 201 peak or residual friction angle values (φp’, φr’) when the stress state of such a point is on the 202 peak or residual yield surface envelopes respectively. 203 A similar analysis of the progressive failure mechanism presented in [31,50] has been carried 204 out. In Fig. 6a the progressive failure phenomenon is represented by plotting the evolution of 205 the MFA (Eq. 5) in 7 material points located along the shear band. It indicates that the 206 degradation of the material initiates at the foot of the slope and propagates upwards. 207 According to Fig. 6a, it is clear that points along the failure surface reach the peak yield 208 envelope at different moments depending on the evolution of the progressive failure 209 mechanism. Note that time is controlled by the evolution of pore pressures because the internal 210 mechanical transfer of stresses in the slope is an instantaneous process. 211 In Fig. 6b, the evolution of the progressive failure is represented in terms of the mean mobilised 212 friction angle. It is obtained by averaging the MFA of 20 material points distributed along the 213 initial shear band and it is a measure of the mean mobilised strength in the failure surface. A 214 very similar behaviour was observed in modelling of the Selborne experiment in [31]. Due to 215 the increase of water pressure in the slope induced by the pressure condition imposed along the 216 bottom boundary (see Fig. 3), the mean MFA increases up to a maximum value. Afterwards, 217 there is a drop of the available mobilised strength. Then, the progressive failure develops, 218 maintaining the mean mobilised friction angle approximately constant. This process ends 219 abruptly at t=8.6 s, when the final point in the failure mechanism reaches the peak condition and 220 immediately afterwards it softens down to the residual state. This leads to the onset of instability 221 and the motion begins. 222 The maximum average mobilized friction angle is attained at t = 8.25 s, when the lower part of 223 the failure surface has already entered into a post-peak strength. This maximum is intermediate 224 between peak and residual strengths and, in the case analyzed, close to the residual value. If a 225 Limit Equilibrium method is used to analyze the slope stability, the maximum calculated at t = 226 8.25 s is reasonable choice for the soil strength. 227 Beyond the maximum the average friction decreases somewhat but the process of progressive 228 failure develops at a fairly constant value of the average friction. When the last point in the 229 failure surface reaches peak conditions there is a sudden reduction in average friction and the 230 slope accelerates. This is indicated in Figures 6b and 7a as the “outset of instability”. Up to this 231 time slope displacements are small and unnoticeable at the displacement scale selected to plot 232 Figure 7a. 233 234 The patterns of displacements, after a sliding mechanism was fully developed, follow the 337 description given when interpreting Figures 6 and 7a. Figure 12a shows the effect of IB on 338 displacements of point P, located at the lower part of the slope. Velocities are also given in 339 Figure 12b. The slide accelerates, reaches a maximum velocity and moves forward towards a 340 new stable profile. 341 Additionally it can be observed that in slopes exhibiting larger values of IB, for the same peak 342 strength: (1) the instability occurs earlier; (2) the velocity increases more suddenly; (3) the peak 343 velocities reach higher values; (4) more time is required to reach the final position at rest; and 344 (5) the run-out is longer. 345 346 (a) 347 348 (b) 349 Fig. 9. (a) Relationship between run-out and IB. (b) Relationship between maximum displacement 350 achieved by a point and IB. All simulations have the same peak strength (cp’=5 kPa and φp’=35º). ∆P 351 indicates the imposed pore water pressure which induced the failure. 352 353 Fig. 10. Final geometry for two simulations with same IB. The displacements of the material points are 354 indicated in the colour scales. Also indicated is the run-out. 355 356 Fig. 11. Final geometries of simulations with cr’=0.5 kPa. The displacements of the material points are 357 indicated in the indicated colour scale. Also indicated is the run-out. 358 359 (a) (b) 360 Fig. 12. (a) Displacement and (b) velocity of the material point P for simulations characterized by cr’=0.5 361 kPa. 362 5.2 Change in peak strength and varying residual friction 363 In the previous Section it was found that a unique relationship developed between run-out and IB 364 when the peak friction strength envelope was constant (and the residual strength varied in a 365 wide range). The next step was to check if such uniqueness would also hold if peak strength 366 parameters change. In order to explore this scenario three different peak strength parameters 367 were selected, rather arbitrarily, but always ensuring that the slope would fail under the imposed 368 water pressure increase at the lower boundary (∆P=70 kPa,): 369 • cp’=5 kPa, φp’=35º (already analysed in the previous section) 370 • cp’=5 kPa, φp’=45º 371 • cp’=9 kPa, φp’=20º 372 For each case, several simulations have been carried out varying the residual strength 373 parameters according to Table 3. 374 Following the procedure described previously, the brittleness index IB has been calculated for 375 each unstable simulation. The effect of IB on run-out is presented in Fig. 13a. Although it is 376 clear that the run-out increases for increasing IB, two different relationships can be 377 distinguished. Whereas the combination of cp’=5 kPa and φp’=45º matches with the correlation 378 defined in Fig. 9a, those simulations with cp’=9 kPa and φp’=20º define higher run-outs. Fig. 379 13b shows the variation of the maximum displacement achieved by a point depending on IB, 380 and, as shown in Fig. 9b, the scatter increases especially for higher values of IB. 381 Since the obtained IB-run-out relationship (Fig. 13) is not unique, three simulations with 382 different peak strengths and the same IB are analysed in detail (Figs. 14 and 15). The evolution 383 of strain contours (Fig. 14) indicates that the shear strains localise along a single band in the 384 first two simulations (Figs. 14a and 14b). By contrast, a deeper mechanism is developed in the 385 third simulation (Fig. 14c) which is characterized by a higher cohesion and a lower friction 386 angle with respect to the other two cases. A deeper seated failure involves a larger volume of the 387 mobilized mass and also a longer length of the sliding surface. It seems that the IB-run-out 388 relationship is also dependent on the failure mechanism. Final displacement fields are given in 389 Fig. 15. Note that values of run-out and maximum displacements are different. 390 Table 3. Run-out and IB for all simulations performed with three peak yield surface envelopes. 391 ∆P=70 kPa cp’=5 kPa φp’=35º cp’=5 kPa φp’=45º cp’=9 kPa φp’=20º cr’ [kPa] φr’[º] IB Run-out [m] IB Run-out [m] IB Run-out [m] 6 20 stable 0.0 6 15 0.3 1.45 6 10 0.42 4.8 6 5 0.56 13.41 5 35 stable 0.0 5 25 stable 0.0 5 20 0.32 1.4 0.25 1.3 5 15 0.43 3.4 0.36 3 5 10 0.54 6 0.48 8.7 5 5 0.63 11.6 0.63 15 5 0 4 30 stable 0.0 4 25 0.28 1.45 4 20 0.39 2.8 4 15 0.48 5.2 2.5 35 0.22 0.7 2.5 30 0.33 2.25 2.5 25 0.41 3.35 2.5 20 0.49 5.26 0.38 5.37 2.5 15 0.57 8.25 0.5 9 2.5 10 0.67 12.7 0.63 13.9 2.5 5 0.75 17 2.5 0 0.87 21.1 1.5 30 0.37 2.92 1.5 25 0.46 4.86 1.5 20 0.56 6.45 0.65 9.3 0.46 8 1.5 15 0.64 10 0.57 12.55 1.5 10 0.73 14.6 0.68 17.02 1.2 25 0.5 5.7 1.2 20 0.6 7.5 1.2 15 0.67 11.7 1.2 10 0.75 15.6 0.5 35 0.39 3.5 0.5 30 0.45 5.4 0.5 25 0.55 6.5 0.65 10 0.5 20 0.63 8.9 0.7 13.1 0.52 12.7 0.5 15 0.7 13.01 0.77 15.4 0.61 15.07 0.5 10 0.79 17.5 0.84 19.5 392 393 (a) (b) 394 Fig. 13. Relationships between IB and (a) run-out and (b) maximum displacement achieved by a point. 395 396 (a) (b) (c) 397 Fig. 14. Distribution of the shear strain at three different times (the initiation of failure mechanism, an 398 intermediate time, and final geometry) for three simulations with similar IB but different peak strength 399 envelopes.(a) cp’=5 kPa, φp’=35º; (b) cp’=5 kPa; φp’=45º; (c) cp’=9 kPa, φp’=20º. 400 401 402 (a) (b) (c) 403 Fig. 15. Final distribution of total displacements field for 3 cases with similar IB but different peak 404 strengths. (a) cp’=5 kPa, φp’=35º; (b) cp’=5 kPa, φp’=45º; (c) cp’=9 kPa, φp’=20º. 405 5.3 Effect of cohesion and friction angle decrease in the onset of failure 406 The onset of failure is analysed depending on the cohesion and friction decrease (Eqs. (10) and 407 (11) respectively) and on the external triggering action (pore water pressure increase in the 408 lower boundary ∆P). Consider the following “brittleness” ratios for effective cohesion and 409 friction: 410 ( ) pr p dc c c c ′ ′′′ = − (10) 411 ( ) ' tan tan tan pr p d ϕ ϕϕϕ ′′′ = − (11) 412 Zero values of these indices corresponds to a ductile behaviour whereas a unit value represents a 413 highly brittle response. 414 All the combinations of dc′ and 'd ϕ shown in Table 2 for ∆P =40 kPa and ∆P=70 kPa are 415 shown in Fig. 16. It is clear that the lower the increments of water pressure, the higher is the 416 required strength reduction to make the slope unstable. For instance, in the case of ∆P =40 kPa, 417 in order to reach failure, the soil should exhibit a full brittleness in one of the strength 418 parameters and a full ductility in the other, or the combination given by the threshold straight 419 line separating failure from stability. 420 These results suggest that both cohesion and friction angle play a similar role in determining a 421 threshold that define whether the slope will become unstable, or on the contrary, will remain 422 stable. 423 424 (a) 425 426 (b) 427 Fig. 16. Stability of the slope depending on the combination of cohesion drop ( dc′ ) and friction angle 428 decrease ( d ϕ ′ ). The increase of pore pressure at at the lower boundary is (a) ∆P =40 kPa and (b) ∆P=70 429 kPa. The same peak strength is maintained in all these simulations (cp’=5 kPa and φp’=35º). 430 5.4 Effect of peak and residual strength in run-out 431 In previous sections, the IB-run-out relationship is analysed but the relevance of peak and 432 residual strength is not discussed. This is because IB combines both effects in a single parameter. 433 The influence of peak and residual strengths on the value of run-out is shown in Fig. 17. It is 434 clear that simulations having the same residual strength have quite similar values of run-out 435 even if different peak yield surface envelopes define the material (Fig. 17). 436 437 Fig. 17. Influence of residual strength on run-out for three different peak Mohr-Coulomb envelopes. 438 6 DISCUSSION 439 6.1 Run-out vs maximum displacement 440 It has been shown that run-out, defined as the distance between the toe of the initial slope and 441 the toe once equilibrium has been re-established after the instability, is not equivalent to the 442 maximum displacement achieved by any point of the slope (see Figs. 9 and 13). While a clear 443 relationship cannot be obtained between IB and maximum displacement, IB and run-out correlate 444 well. 445 The difference between run-out and maximum displacement is evident especially when the 446 failure mechanism is deep and the landslide is essentially a rotational movement (Fig. 10a). The 447 deeper the failure surface (cohesive component of strength dominates) the larger the ratio 448 between run-out and maximum point displacements. However, both lengths are similar when 449 the initial failure is shallow (Fig. 10b). 450 6.2 Effect of peak and residual strength in the whole instability process 451 Here the role played by peak and residual strengths in the stability of the slope, in the slip 452 surface geometry and in the post-failure response is discussed. 453 According to the results presented in Fig. 6, it is clear that the peak envelope controls the 454 initiation of the progressive failure because it determines when the first point reaches the 455 maximum strength. However, the redistribution of stresses due to the strain softening of the 456 material and the propagation of the progressive failure is a complex process governed by both 457 peak and residual states. Note that the mean mobilised strength in the slope (Fig. 6b) remains 458 always below the peak value. 459 In agreement with this, it has been observed that the geometry of the failure mechanism is 460 definitely influenced by both peak and residual strengths (Figs. 8 and 14) but peak strength has 461 a stronger effect. Especially the peak cohesion highly influences the depth of the mechanism. 462 Finally, the run-out is essentially influenced by the residual state (Fig. 17). It makes sense 463 because when the post-failure stage initiates the soil in the shear band has experienced enough 464 plastic shear strain to be totally softened. This behaviour is also shown in Fig. 6b. 465 6.3 Residual cohesion in brittle soils 466 In brittle soils, peak friction angles may take values ranging from 5º to 45º depending on the 467 type of soil. The variability of peak cohesion can be also very large (from 0 kPa to more than, 468 say, 200 kPa in very stiff clays). However, residual effective cohesion is very low or non-469 existent. 470 The selection of peak and strength values presented in the parametric analysis (Section 5) 471 represents a large variability of strain softening materials, and some of them include unlikely 472 values for the residual cohesion (up to 6 kPa). 473 In order to analyse if this restriction have some effect on the results, an additional figure is 474 included here (Fig. 18) in which only those simulations from Table 3 having a small cr’ (cr’≤1.5 475 kPa) are presented. The number of cases in the simulations performed decrease substantially but 476 the relationships between IB and run-out and maximum displacement look essentially the same 477 as those obtained when interpreting the complete set of simulations (Fig. 13). 478 479 (a) (b) 480 Fig. 18. Relationships between IB and (a) run-out and (b) maximum displacement achieved by a point. 481 Cases with cr’≤ 1.5 kPa. 482 7 CONCLUSIONS 483 The stability and post-failure behaviour of a saturated slope have been analysed by means the 484 MPM which has been proved that it is capable to simulate both the initiation of failure, which 485 involves small strains, and the post-failure stage, generally characterised by large displacements. 486 A homogeneous slope with a regular geometry has been analysed. The slope failure is triggered 487 by increasing the water pressure on the lower boundary of the domain. 488 The slope material has been defined by a strain-softening elastoplastic constitutive law that 489 allows the simulation of the strength decrease (from a peak to a residual value). The progressive 490 failure mechanism, typically observed in brittle materials, is reproduced and analysed. Both, 491 peak and residual values of the strength control the slope failure which progresses from the toe 492 to the crest of the slope. On the contrary, the post-failure behaviour is mainly controlled by the 493 residual strength and it has an important effect on the run-out. The results show that the 494 geometry of the failure surface determines the final displacement field. 495 The effect of the material brittleness, defined in terms of brittleness index IB (proposed by 496 Bishop [12]), on the post-failure behaviour has been identified by means of a parametric 497 analysis combining peak and residual values of cohesion and friction angle. Both run-out and 498 the maximum displacements have been represented in terms of the brittleness index. Run-out 499 was defined here as the distance between the toe of the initial slope and the toe of the slope after 500 failure once equilibrium has been re-established. It was found that run-out increases with IB and 501 both correlate well when a common peak strength envelope is adopted. On the contrary, a clear 502 relationship has not been obtained between IB and maximum displacement. 503 The onset of failure also depends on the magnitude of the triggering mechanism. The higher the 504 intensity of the triggering mechanism, the lower IB is sufficient to induce instability. This fact 505 allows defining a brittleness threshold ˆ P B I ∆ which determines the minimum brittleness required 506 to induce instability for a certain excess pore pressure. However, the magnitude of the applied 507 excess pressure (∆P) does not change the general observations discussed above. 508 8 ACKNOWLEGDMENTS 509 The software used in this work is a version of the MPM code developed by the MPM Research 510 Community formed by 6 partners: Deltares (The Netherlands), Universitat Politècnica de 511 Catalunya (Spain), University of Cambridge (UK), Technische Universität Hamburg-Harburg 512 (Germany), Università degli Studi di Padova (Italy), and Delft University of Technology (The 513 Netherlands). 514 9 REFERENCES 515 [1] Fell R, Hungr O, Leroueil S, Riemer W. Keynote lecture - Geotechnical engineering of 516 the stability of natural slopes, and cuts and fills in soil. ISRM Int. Symp., Melbourne, 517 Australia: International Society for Rock Mechanics; 2000. 518 [2] Pinyol N, Alonso E, Corominas J, Moya J. Canelles landslide: modelling rapid 519 drawdown and fast potential sliding. Landslides 2012;9:33–51. 520 [3] Alonso E, Gens A. 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