Francesc Mirada, Arup

## Abstract

The material point method (MPM) is an advanced particle-based method that combines Eulerian and Lagrangian descriptions of the media to describe the dynamic behaviour of the continuum. The novelty of the method is the capability to deal with large deformations avoiding problems of mesh tangling typical from finite elements. In recent years, it has been extended to solve problems in soil mechanics. MPM has a good potential to examine the conditions leading to slope failure but, also, it is capable of following in time the evolution of the unstable mass determining its final runout, which is a key variable to evaluate the consequences of instability. The aim of this work is to contribute in the validation of the method and prove its functionality by modelling a real slope failure. In this work, the MPM code Anura3D is used to analyse the Selborne experiment, which was a landslide induced by water recharge in saturated soils. Moreover, a novel feature recently implemented to simulate excavation processes is used to study the influence of the initial stresses on the generated slip surface and the post-failure behaviour. The Mohr-Coulomb model with strain softening is considered to simulate the soil brittleness. Numerical results show the evolution of a progressive failure and final displacements similar to those observed in the field.

## Introduction

The Material Point Method (MPM) is a numerical technique to model large deformations. It combines advantages of both, the Eulerian and Lagrangian despcriptions to tackle large deformation problems. Many problems in geomechanics involve large deformations and large movement of soil masses such as landslides, pile penetration, tunnels or tunnel collapses.

Landslide analysis is also an essential part for risk assessment and these methods are an emerging tool to predict the character of the failure and give a quantitative estimation of the post failure behaviour, including the travel distance (run-out) and the velocity.

## Basis of the MPM

The MPM is a numerical technique to model large deformations combining the particle-in-cell methods and Finite Element Methods (FEM) [^{1, 2}]. In the MPM, the continuum is modelled by means a set of Lagrangian points, so-called material points or particles. The material points carry all physical variables in order to define the state of the material such as mass, material parameters, strains, external loads, etc. Moreover, an Eulerian mesh, which is fixed through the calculation, is used as a computational mesh to solve the governing equations of the motion. The information from the material points to the computational mesh is transferred at the beginning of every time step. At the end of the step, the information is mapped again from the mesh to the material points in order to update the information. Figure 1 shows the concept of how the method works.

Through this approach, MPM combines the advantages of both Lagrangian and Eulerian formulations. It avoids the Eulerian problem of the convective term which generates the numerical diffusion. In addition, it solves the problem of the mesh distortion associated to the Lagrangian mesh working on large deformations.

The code used is a version of Anura3D developed by the MPM Research community. Within this work a hydromechanical fully coupled dynamic formulation is used [^{3}].

## The Selborne experiment

The experiment consisted of a 9m deep cut slope in Gault Clay which was induced to failure by a pore pressure increase. The main goal was to study the progressive failure mechanisms generated. For that reason, the site was extensively instrumented by means of piezometers, inclinometers and surface wire extensometer lines. Field sighting and measurements are reported in [^{4-6}].

## The MPM model

In previous works [^{7, 8}], the Selborne experiment was modelled by using the MPM. In these analyses, numerical parameters of the soil were chosen according to field data, and the water recharge was modelled in agreement with piezometer’s measurements. The progressive failure was simulated and the results showed a final run-out very similar to that observed in the field. However, in those analyses, the excavation process was not simulated and the material was supposed to be normally consolidated with K_{0}= 0.5.

The novelty of this work is that the slope cutting process is incorporated in the modelling. The effect of the initial stress field, essential to reproduce the failure progression more accurately, will be analysed. Initially, a K_{0} value of 2.0 is taken to generate the initial stresses. This value has been considered taking into account field measurements [^{5}] in the overconsolidated clay layers. Geometry, material parameters, and loading conditions have been taken from previous analysis [^{7, 8}].

The simplified geometry of the Selborne experiment consists on a slope of 9m high and 63º steep (see figure 2).

An excess pore pressure is applied along 24 m of the bottom boundary in order to simulate the water recharge. It is linearly increased during 10 seconds from 0 to 110kPa. Afterwards, the excess pressure on the lower boundary is kept constant.

Figure 3 shows the computational mesh with tetrahedral elements and initial distribution of material points used for the calculations.

The strain softening Mohr-Coulomb constitutive law presented in [^{9}] is used to model the brittle behaviour of the soil. The material properties assumed while simulating the Selborne experiment are presented in Tables 1 and 2.

### Table 1: Soil parameters used in the simulation

Soil parameters | |
---|---|

Porosity | 0.3 |

Intrinsic permability (m^{2}) |
10^{-10} |

Density of the solid (kg/m^{3}) |
2,700 |

Youngs modulus (kPa) | 20,000 |

Poisson coefficient | 0.33 |

Calibration parameter for the MC-SS | 500 |

### Table 2: Shear strength parameters for the soils

Material | Peak shear Strengthc’ [kPa] Φ’ [kPa] | Softened Shear Strengthc’ [kPa] Φ’ [kPa] | ||
---|---|---|---|---|

Weathered Gault Clay |
13 |
24.5 |
4.0 |
13.5 |

Gault Clay |
25 |
26 |
0.5 |
15 |

## Results

Figure 4(a) shows the localisation of deviatoric strain just when the slope failure mechanism is initialised. The slip surface obtained in this analysis is compared in figure 4(c) with the one observed in the field, and the one obtained in previous MPM analysis in which the excavation process was not simulated (figure 4(b)). It is shown that considering the initialisation of stresses with a more realistic K_{0}, and the modelling of the excavation process the slip failure fits much better the real failure mechanism.

For a more detailed analysis of the progressive failure mechanism, the mobilised shear strength concept is used. The mobilised shear strength concept was introduced in [^{10}] and is a measure of the intensity of shear in a certain point. The concept of mobilised shear friction angle, sin φ’, is defined in equation 1.

Equation 1

Note that under peak and residual conditions φ’ coincides with the peak and residual friction angles respectively.

In Figure 5, the progression of the local shear failure along the developing slip surface is shown according to field measurements and observations made in [^{6}]. The concept of the progression can be studied quantitatively with the mobilised shear strength concept. It can also be seen that at a very early stage, the two inclinometers located in the toe of the slope (I.09 and I.09) already reached peak strength values, even before the start of the pore pressure increase. Then, the points located in the crest of the slope start plastifying. Finally, the failure progress from the toe and the crest to the centre of the slope until the failure mechanism is completed. Immediately afterwards, the global instability begins.

For the sake of comparison with field measurements the time is normalized (*t**) with the elapsed time from the initiation of failure until final stabilisation of the landslide Three different intervals can be distinguished, which are described in Table 3.

### Table 3: Description of dimensionless time intervals (*t**)

Non-dimensional interval | Meaning |
---|---|

From -1 to 0 |
Time period corresponding to the excavation process and generation of the initial stresses |

From 0 to 1 |
Time elapsed between the initiation of pore pressure increase and the completion of the progressive failure mechanism. Only small deformations appear. |

From 1 to infinity |
Time elapsed between the initiation of the global slope failure and the final equilibrium. Large deformations appear. |

Figure 6(a) illustrates the evolution in time of the mobilised shear angle for seven representative points located along the shear band. Note that dimensionless time (*t**) comprehended between -1 and 0 is the time taken for the excavation before the pore pressure increase starts. It can be seen that the points located in the toe of the slope reach the peak shear strength values before the initiation of the water recharge: first the point located at the toe and then the deeper ones, which are the points P7, P6 and P5, respectively. At *t** = 0, just when the water pressure in the slope increases, points P7 and P6 have already reached the residual strength parameters.

In Figure 7(b), the mobilised friction angles during the water recharge are presented. The first material point to reach the peak strength values is P1, located in the crest of the slope, followed by P2, due to the stress redistribution in that area at *t**=0.42 and 0.43 respectively. After that, P3 and P4 reach the peak strength values at *t**=0.58 and 0.73 respectively. It can be observed that at *t**=0.93 all the points in the shear band are in residual strength conditions, except for the point P4 that is still decreasing its mobilised friction angle. Finally at *t**=1.0 all the points had reach the residual yield surface, the failure mechanism is completed and the global instability is initiated.

## Influence of the initial stresses

Using the same geometry and material properties of the Selborne experiment a parametric study of the K_{0 }values has been performed to understand the influence of the initial stresses on the slip surface geometry and the post-failure behaviour.

## Slip surface

In Figure 8, there are presented and superposed all shear bands observed for all instable cases. It can be observed that whereas K_{0} is increased the slip surface is deeper in the crest of the slope. The slip surface in the toe is coincident for each of the cases. Note that the slip surface when K_{0}=1.0 starts at the toe but it does not extend to the rest of the slope. A K_{0} value of 1.0 locates the stress state of the material points in the isotropic axis of the *p’-q* plane, and then a higher deviatoric stress is required in order to cause the failure. The pore pressure increase applied in this analysis (110kPa) is not enough to trigger the failure. For this reason, the slope remains stable for this configuration.

## Kinematic effect

In figure 9 (note the different scales) there are presented the final horizontal displacements field for each K_{0} value. It can be observed that if the instability occurs, as the K_{0} value of the soil increases the final run-out of the soil mass decreases. The run-out for the non-excavated case is the largest one.

The evolution of horizontal displacement of a material point initially located in the middle of the mobilised soil mass above the slip surface are shown in Figure 10. It can be seen that as mentioned above, the more over-consolidated is the soil, the less horizontal displacements are observed.

It is important to note that the smaller is the overconsolidated ratio the less time is required to reach the stable geometry (see figure 10).

## Conclusions

Within this work, the Selborne cutting experiment has been modelled. The numerical results show that the initial failure mechanism obtained in this analysis is much more similar to the real one than the one presented in previous works [^{7}]. This fact leads to conclude that the initial stress state highly influences the geometry of the slip surface and the postfailure behaviour.

Therefore, it is essential to model as accurately as possible the soil history; in this case by considering the over-consolidated state of the clay layers and the unloading process due to the cutting of the slope. Moreover, the progressive failure mechanism has been correctly reproduced. The first area to reach the peak strength values is the toe of the slope, even before the beginning of the pore pressure increase, due to the excavation effect.

Afterwards, when the pore pressure surcharge initiates, the crest area of the slope plastifies and the failure propagates towards the centre of the slope. Finally, the progressive failure is extended to the rest of the slope and the global failure occurs. Those three phases of the progressive failure matched the real results observed in the site.

Additionally, using the same geometry, loading conditions and material parameters, an extended parametric analysis has been carried out in order to study the influence of the initial stresses in the soil. Different K0 values are considered. The results showed that the initial stresses value does not only affect the failure process but also the post-failure behaviour of the mobilised soil mass. The higher the K_{0}, the deeper the initial slip surface. Besides, different pattern for the different K_{0} values are observed. For lower K_{0} values a rapid increase of the velocity is observed and then stabilized whereas for higher K_{0} values the process is affected by the stability-reactivation effect where subsequent instabilities stages occur. For more over-consolidated soils, the final displacements and the velocities achieved are smaller than for the less overconsolidated soils.

This current work contributed to further validate the MPM, and highlighted the potential of the method which is not only capable of describing the conditions leading to slope failure but also, it is capable of following in time the post-failure behaviour of the unstable mass of soil. Moreover, this work highlighted the importance of the initial stress state modelling in order to approach the correct results in slope stability analysis.

## Acknowledgements

Results from the Selborne experiment [^{4-6}] are used with the permission of the authors. The author wants to thank Eduardo Alonso from Polytechnic University of Catalonia, Alba Yerro from Cambridge University, Alex Rohe and Mario Martinelli from Deltares for their support and advice in this work.

## References

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