# The role of constitutive models in MPM simulations of granular column collapses

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## Abstract

The granular column collapse is a well-established experiment which consists of having a vertical column of granular material on a flat surface and letting it collapse by gravity. Despite its simplicity in execution, the numerical modelling of a column collapse remains challenging. So far, much attention has been dedicated in assessing the ability of various numerical methods in modelling the large deformation and little to the role of the constitutive model on both the triggering mechanism and the flow behaviour. Furthermore, the influence of the initial density, and its associated dilatancy and strength characteristics, have never been included in the analyses. Most past numerical investigations had relied on simple constitutive relations which do not consider the softening behaviours. The aim of this study is to illustrate the influence of the constitutive model on the on-set of failure, the flow behaviour and the deposition profile using the material point method. Three constitutive models were used to simulate the collapse of two granular columns with different geometries and for two densities. The results of the simulations showed that the constitutive model had a twofold influence on the collapse behaviour. It defined the volume of the mobilised mass which spread along the flat surface and controlled the dissipation of its energy. The initial density was found to enhance the failure angle and flow behaviours and was more significant for small columns than for larger ones. The analysis of the potential energy of the mobilised mass explained the existence of two collapse regimes.

## Keywords

Column collapse Constitutive modelling Dilatancy Granular flow Large deformation modelling Material point method## 1 Introduction

*Regime 1*) concerned columns with small aspect ratios and the inertia effect was limited. A small volume of mass was mobilised and slid in a single flow motion. Two sub-categories were proposed by Lajeunesse et al. [18] depending upon whether the deposition was truncated (

*Regime 1a*) or totally consumed by the collapse (

*Regime 1b*). The second type (

*Regime 2*) concerned taller columns in which the inertia effect dominated the collapse behaviour and resulted in complex multi-flow behaviours. The deposition profile took the shape of a ‘Mexican hat’. Balmorth and Kerswell [2] pointed out that there was a gradual transition from the ‘slow avalanches of shallow columns’ (

*Regime 1*) to ‘violent cascading collapses of tall columns’ (

*Regime 2*).

*a*is the aspect ratio, \(h_0\) the initial height and \(r_0\) the initial radius.

## 2 Simulating the column collapse

Despite the apparent simplicity of the experiment, the description and prediction of the collapse remains challenging from an experimental, numerical and theoretical point of view [29]. Many attempts to model the column collapse have already been undertaken both with particle and continuum based methods.

Staron and Hinch [36] presented discrete element (DEM) simulations which showed good agreement with the experimental results in terms of run-out distance. However, they commented on the absence of physical explanation on the power law relationship between the initial aspect ratio and the run-out distance. Furthermore, the influence of the material properties and the initial state on the collapse behaviour were not investigated. Zenit [44] also presented DEM simulations using soft particles and observed differences between the simulations and the experimental results which were attributed to the estimation of the angle of repose. Both Staron and Hinch [36] and Zenit [44] focused on the final deposition profiles with little insight on the collapse mechanism. Lacaze et al. [17] carried out DEM simulations with good agreement with the experimental results, both in terms of flow behaviour and run-out distance. However, the authors commented on the necessity of investigating the influence of multi-sized particles. Girolami et al. [13] used spheres rather than discs in their DEM simulation which gave better prediction of the experiments. Utili et al. [40] used multi-sized particles which and gave better results when using an angle of repose closer to experimental values. They commented on the influence of the shape of the grains on the angle repose and discussed the dilatancy characteristics of the granular material but did not consider it in the simulations. Kumar [16] carried out DEM simulations to investigate the role of the initial grain properties and showed that it had had a significant influence on the flow kinematics and the internal flow structure.

DEM is well suited for micro-mechanical analyses but suffers from its computational cost when applied to large scale problems. For this reason, many researchers have favoured continuum methods such as the adaptive Lagrangian–Eulerian finite element method (ALE FEM). Crosta et al. [9] presented a series of simulation using ALE FEM with a Mohr–Coulomb model. The results were in line with the experimental results. However, the authors commented on the computational cost of the method. The smoothed particle hydrodynamics (SPH) method is computationally cheaper when modelling large deformation problems. Chen and Qiu [7] as well as Liang and He [20] carried out simulations using SPH with, respectively, a Drucker–Prager and a rate dependent Mohr–Coulomb failure criteria. Despite good agreement with the experimental data in terms of run-out distances and final heights, both models are simple failure criteria which do not take the density or softening behaviours into account. It is known that softening behaviours play a key role in slope instabilities and that simple models cannot capture the complexity of the mechanical behaviour [28]. Furthermore, SPH suffers from difficulties in applying boundary conditions due to the absence of a computational mesh.

An alternative to SPH is the material point method (MPM) which was developed from the particle-in-cell method (PIC) by Sulsky et al. [37, 38]. MPM is an Eulerian–Lagrangian method designed for large deformation problems. It differs from PIC in that it is formulated in the weak form. This implies that history-dependent constitutive models can be formulated within the material points in the same way as for the finite element method (FEM). MPM can be seen as an ALE FEM in which all computational variables, including mass, are stored in every single material point. Its application to geotechnical engineering has been discussed and demonstrated by Solowski and Sloan [35]. Its ability to tackle fluid-like behaviours of granular material has been demonstrated by Wieckowski [42]. Andersen [1] showed that MPM was able to model the column collapse using a simple Mohr–Coulomb model. Bandara [3] simulated the column collapse with both SPH and MPM and obtained the same results. However, the SPH simulation required a large number of particles to obtain an accurate run-out distance making it computationally more expensive. Solowski and Sloan [34, 35] compared MPM simulations with the experimental data of Lube et al. [23] and showed that the Mohr–Coulomb model did not dissipate sufficient energy. Hence, the run-out distances were largely overestimated and numerical damping had to be applied in order to match the experimental results. Kumar [16] carried out simulations of the column collapse with both MPM and DEM and showed that MPM with a Mohr–Coulomb model suffered from insufficient dissipation of energy in comparison with DEM. It was attributed to the absence of inter-particle collisions which dissipates some energy. He also compared the standard MPM formulation [37, 38] with the generalised interpolation material point method (GIMP) [4] and found no apparent improvement for column collapse simulations.

So far, all the discussions focused on the method itself with little investigation on what role the constitutive model played in the prediction. Mast et al. [24] carried out column collapse simulations using a Drucker–Prager model and a hardening–softening Matsuoka–Nakai model and showed that the choice of the constitutive model impacted the final deposition profile in terms of final height and run-out distance. Furthermore, they showed that an enhancement of the peak strength resulted in larger final heights and shorter run-out distances. Following this path, this paper investigates the role of the constitutive model in the column collapse.

### 2.1 The material point method

### 2.2 Constitutive modelling for large deformation

MPM is used for large deformation simulations in which some regions of the continuum body undergoes large deformations and others small deformations. Therefore, the constitutive model must be able to handle both cases. Many MPM simulations presented in the literature used simple failure criteria such as Mohr–Coulomb (e.g. [1, 3, 16, 34, 35]). The model parameters were chosen as being close to the critical state values (\(\varphi ^{\prime } \approx \varphi ^{\prime }_{\mathrm{cs}}\) and \(\psi \approx 0\)) favouring the large strained areas and neglecting their mechanical behaviour at small strains. The necessity of using more advanced models has already been highlighted in the literature. For instance, Yerro et al. [43] suggested using a Mohr–Coulomb Strain Softening model to simulate progressive landslides and Mast et al. [24] suggested a hardening and softening Matsuoka–Nakai model to simulate column collapses. The aim of this paper is to discuss the role of the constitutive model in simulation of column collapses and to capture the necessary feature of the constitutive model for large deformation modelling.

#### 2.2.1 Critical state constitutive models

*critical state*. At this point, the soil will be continuously deformed without any changes in volume or stresses (Eq. 2). The critical state is uniquely defined in a (

*p*,

*q*,

*e*) space by the critical state locus (CSL) at which point the dilatancy

*D*is nil (Eq. 4) and the stress ratio \(\eta\) constant (Eq. 3).

*q*is the deviatoric stress, \(\varepsilon _v\) and \(\varepsilon _d\) are, respectively, the volumetric and deviatoric strains.

*M*the critical state stress ratio, \(D = d\varepsilon _v / d\varepsilon _d\) the dilatancy rate and \(e_c\) the critical state void ratio.

In this study, it is assumed that the granular material, which has failed and flows, will reach the critical state. However, different soil models will reach this state differently. The critical state can be achieved by classical failure criteria such as Mohr–Coulomb by carefully choosing the model parameters (i.e. critical state friction angle with zero dilation angle). In other cases, the critical state is systematically reached and independently from the choice of the model parameters. These models are referred to as critical state models set within the critical state framework [33]. The two conditions (Eqs. 3, 4) can be simultaneously fulfilled such as in Cam-Clay [30] or independently fulfilled such as in Nor-Sand [14].

#### 2.2.2 Mohr–Coulomb

*c*and the friction angle \(\varphi ^{\prime }\). A yield function can be formulated from it (Eq. 5). It is often implemented with a non-associative flow rule and a potential function given in Eq. 6. It introduces a third parameter—the dilatancy angle \(\psi\). Granular materials are cohesionless (\(c^{\prime } = 0\)) which reduces the number of plastic parameters to two—the friction angle \(\varphi ^{\prime }\) and the dilatancy angle \(\psi\). In this study, the Mohr–Coulomb model was implemented as an elastic–plastic model in which the hardening phase is purely elastic and defined by Young’s modulus

*E*and the Poisson ratio.

*F*is the yield function, \(c^{\prime }\) the effective cohesion, \(\varphi ^{\prime }\) the effective friction angle, \(\theta\) the Lode angle,

*P*the potential function and \(a_{pp}\) the distance to the apex.

#### 2.2.3 Mohr–Coulomb Strain Softening

The reduction in strength and dilatancy allows the model to soften. Following the critical state theory (Eqs. 3, 4), the residual values must be those of the critical state; the residual dilatancy angle must be nil (\(\psi _{\mathrm{res}} = \psi _{\mathrm{cs}} = 0\)) and the residual friction angle must be the critical state friction angle (\(\varphi ^{\prime }_{\mathrm{res}} = \varphi ^{\prime }_{\mathrm{cs}}\)).

The accumulated plastic deviatoric strain \(E_d^p\) is a material point variable stored in the material point and specific to it. It tracks the history of shearing and dictates how it should soften independently of the stress state and density of the soil. Mohr–Coulomb Strain Softening possesses some deficiencies. The peak strength is modelled as a yielding point, and therefore, the hardening phase is a purely elastic behaviour. Furthermore, the peak strength occurs at the end of the contraction phase and before any dilatancy take place. The model ignores the fact that the peak strength of a granular material is determined by its dilatancy characteristics. Taylor [39], followed by Rowe [32] among others, showed that the peak strength was the sum of the critical state strength and the maximum dilatancy rate and is known as the stress-dilatancy theory. The maximum dilatancy rate is density and pressure dependent [5, 6]. Therefore, the dependency on the density is implicitly embedded in the model parameters of Mohr–Coulomb (friction and dilatancy angle in this case).

#### 2.2.4 Nor-Sand

*M*the critical state stress ratio,

*N*the dilatancy parameter and \(p_i\) is the image pressure.

*Q*which is the pressure at which individual soil particles are broken apart [6].

#### 2.2.5 Calibration of the constitutive models

Mohr–Coulomb parameters

Parameter | Symbol | Value |
---|---|---|

Young’s modulus | | 10 MPa |

Poisson ratio | \(\nu\) | 0.2 |

Cohesion | \(c^{\prime }\) | 0 kPa |

Friction angle | \(\varphi ^{\prime }\) | \(33^\circ\) |

Dilatancy angle | \(\psi\) | \(0^\circ\) |

Mohr–Coulomb Strain Softening model parameters

Parameter | Symbol | Value | |
---|---|---|---|

loose | dense | ||

Young’s modulus | | 10 MPa | 10 MPa |

Poisson ratio | \(\nu\) | 0.2 | 0.2 |

Peak cohesion | \(c_{\mathrm{peak}}\) | 0 kPa | 0 kPa |

Residual cohesion | \(c_{\mathrm{res}}\) | 0 kPa | 0 kPa |

Peak friction angle | \(\varphi ^{\prime }_{\mathrm{peak}}\) | \(39^\circ\) | \(50^\circ\) |

Residual friction angle | \(\varphi ^{\prime }_{\mathrm{res}}\) | 33\(^\circ\) | \(33^\circ\) |

Maximum dilatancy angle | \(\psi _{\mathrm{max}}\) | \(6^\circ\) | \(25^\circ\) |

Residual dilatancy angle | \(\psi _{\mathrm{res}}\) | \(0^\circ\) | \(0^\circ\) |

Shape function | \(\beta\) | 4 | 5 |

Nor-Sand model parameters

Parameter | Symbol | Value |
---|---|---|

Shear modulus constant | | 2500 |

Shear modulus exponent | | 0.5 |

Poisson ratio | \(\nu\) | 0.2 |

Triaxial compression critical state stress ratio | \(M_{\mathrm{tc}}\) | 1.33 |

Dilatancy parameter | | 0.3 |

Maximum void ratio | \(e_{\mathrm{max}}\) | 0.946 |

Minimum void ratio | \(e_{\mathrm{min}}\) | 0.500 |

Crushing pressure | | 10 MPa |

Triaxial compression dilatancy coefficient | \(\chi _{\mathrm{tc}}\) | 3.5 |

Minimum hardening modulus | \(H_{\mathrm{min}}\) | 25 |

Hardening coefficient for compression | \(\delta _H\) | 1850 |

### 2.3 Definition of geometry and mesh

*Column 1*) and 2.0 (

*Column 2*) as shown in Fig. 7. These two initial aspect ratios are respectively in the upper limit of

*Regime 1*and lower limit of

*Regime 2*according to the experimental data [18, 19, 21, 22, 23]. The columns rested on a thin layer called the base layer which provided the friction necessary for the deposition. It is modelled as a stiff elastic body. According to the experimental data, the friction of the base layer plays a small role in the column collapse [18] and was confirmed numerically when some realistic friction angles were applied [3]. The opening of the gate was not modelled as such in the simulation. It was assumed that it was instantaneous and had no affect on the collapse mechanism. However, this may lead to some differences between the experimental and numerical observations, which was not investigated in this study.

MPM model setup information of the two column geometries

Column | Size | MP/cell | MPs | Cells | Nodes |
---|---|---|---|---|---|

1 | 0.8 | 4 | 5196 | 3672 | 7653 |

1 | 0.8 | 10 | 12,990 | 3672 | 7653 |

1 | 0.6 | 4 | 8808 | 6300 | 12,999 |

1 | 0.4 | 4 | 20,256 | 14,370 | 29,259 |

2 | 0.6 | 4 | 10,392 | 13,788 | 28,179 |

The MPM code used for this study was provided by the MPM Research Community (http://mpm-dredge.eu). The domain was meshed with unstructured tetrahedral elements and is used to initialise the material points. It implied that fine meshes with smaller cells initialised more material points with smaller masses than with coarser meshes. The unstructured nature of the mesh implied that the mass of a material point can differ from one point to another as the cells have different volumes. However, for the present simulations, the mesh was very regular and extreme differences in cell sizes were avoided. The number of material points initialised in each cell can be changed. A mesh sensitivity analysis was carried out prior to the analysis in order to understand and minimise its influence. Three mesh sizes were investigated for which the number of material points per cell ranged from 4 to 10. Table 4 summarises the different cases.

The results of the mesh sensitivity analysis are shown in Fig. 8 in which the mesh is visible. The colour mapping represents the deviatoric strain—0 % blue and 100 % red. The results show that the failure surface, which is the interface between the blue and the red material points, was influenced by the mesh size. It is concave for the coarse meshes and convex for the fine meshes. It will be shown later that the convex shape is related to a process called avalanching which is mesh-dependent. Increasing the number of material points in a coarse mesh did not significantly improve the results as was also observed by Kumar [16]. However, a refinement of the mesh, which indirectly initialised more material points, improved the results significantly.

## 3 Simulation results for *Column 1*

*Column 1*are presented in Fig. 9. Two snapshots are shown for each constitutive model and density. The first snapshot is at \(T = 0.3\) s. It shows the primary failure surface and the formation of the sliding wedge. The second snapshot is at \(T = 2.5\) s and shows the final deposition profile. The colour mapping corresponds to the deviatoric strain— 0 % blue and 100 % red. Therefore, the static cone is mapped in blue and the mobilised mass in red.

The simulations with the Mohr–Coulomb Strain Softening model were carried out for two initial densities (loose and dense) with two different sets of model parameters (loose: \(\varphi ^{\prime }_{\mathrm{peak}} = 39^\circ\), \(\psi _{\mathrm{peak}} = 6^\circ\), dense: \(\varphi ^{\prime }_{\mathrm{peak}} = 50^\circ\), \(\psi _{\mathrm{peak}} = 25^\circ\)). The results for the loose case show a fast developing failure surface which defines the static cone and the mobilised mass. The angle of the failure surface is steeper than for the Mohr–Coulomb simulation with critical state parameters and is due to the higher peak strength. Consequently, the initially mobilised mass is smaller and the static cone larger. During the collapse, the mobilised mass slides as a rigid body along the failure surface and crumbles upon contact with the base. Figure 9c shows the collapse at \(T = 0.3\) s in which the sliding wedge has started to crumble upon contact with the base. The flow is then progressively slowed down by frictional contact with static layers (\(T = 0.6-0.8\) s). The avalanching process then gradually takes place, eroding the summit of the column (\(T = 0.8-2.5\) s). The deposition profile (\(T = 2.5\)), as shown in Fig. 9d, has a similar shape than for the critical state Mohr–Coulomb but with a shorter run-out distance (\(r^* = 1.5\)). This is due to the smaller mobilised mass and additional dissipation of energy, albeit limited as the mobilised mass has mostly the critical state friction angle. The collapse behaviour of the loose sand with a Mohr–Coulomb Strain Softening model is very similar to the one observed with Mohr–Coulomb. The results for the dense case still show a fast development of the primary failure surface. However, the failure angle is significantly larger and the mobilised mass smaller than for the previous cases as shown in Fig. 9e. It is due to a high peak friction angle. Then, the mobilised mass slides down and crumbles upon contact with the base layer (\(T = 0.4\) s). Static layers are then built bottom-up by frictional contact with the base layer (\(T = 0.6\) s). The high peak friction angle dissipates more energy than for the previous cases, and the run-out distance is shorter (\(r^* = 1.20\)). It is within the range of the experimental predictions but with a steeper deposition slope. However, there is evidence that the experiments were conducted with loose to medium-dense sand rather than dense.

The simulations with Nor-Sand were carried out for two initial densities (loose and dense) but with a unique set of model parameters as Nor-Sand includes the void ratio as a model variable. The results for the loose case show that the development of the primary failure surface is slower than for the Mohr–Coulomb cases. A sensitivity analysis showed that this lag time is caused by the plastic hardening. The position and angle of the failure surface evolves with the plastic hardening. Figure 9g shows the collapse at \(T = 0.3\) s and in which the ‘hardened’ failure surface is shown. The mobilised mass formed a wedge which slide along the failure and crumbled upon contact with the base (\(T = 0.5\) s). It then flowed as a single mass. The material in contact with the failure surface and the base, albeit not exclusively, was slowed down by frictional contact. Successive static layers were gradually build from the base to the surface (\(T = 0.7 - 0.9\) s). No avalanching was observed during the collapse of the column with loose sand. The deposition profile (\(T = 2.5\) s) is a truncated cone with a slope corresponding to the critical state friction angle. The run-out distance is within the range of the experimental prediction (\(r^* = 1.2\)) and is shorter than with the Mohr–Coulomb cases but is in the range of the experimental predictions. It is due to additional dissipation of energy during the hardening phase and the dissipation of energy within the mobilised mass. This point will be further developed in the discussion section.

The results for the dense case with Nor-Sand show a steep failure surface. The speed of the development of the failure surface is influenced by the hardening rate and the failure slope by the peak strength as for the Mohr–Coulomb case. Figure 9i shows the collapse at \(T = 0.3\) s. It can be seen that the failed mass, mapped in red, is smaller than for the loose case. The dilative nature of dense sand causes the sand to expand and explains the way the lateral free surface has underwent high shearing. As the material hardens, the slope of the failure surface increases. Therefore, the mobilised and flowing mass is reduced. Once this reduced mobilised has reached the base and stabilised, an intensive avalanching phase starts (\(T = 0.5\) s) in which the static cone continuously shreds (\(T = 0.7 - 0.9\) s). This process is slow and was still continuing at \(T = 2.5\) s as shown in Fig. 9j. However, the shredded material affected the upper part of the column but did not affect the short run-out distance (\(r^* = 0.75\)). It has been shown by Darve et al. [11] that the avalanching process is strongly related to a diffuse mode of failure and that localised mode (typically by shear band formation) and diffuse mode (typically by avalanches) can coexist spatially and/or appear successively temporally in boundary value problems involving granular media.

## 4 Simulation results for *Column 2*

*Column 2*are presented in Fig. 9. As for

*Column 1*, two snapshots are shown for each simulation, respectively, for constitutive model and density. The first snapshot is at \(T = 0.3\) s which shows the primary failure surface and the formation of the sliding wedge. The second snapshot is at \(T = 2.5\) s which shows the final deposition profile. The colour mapping corresponds to the deviatoric strain—0 % blue and 100 % red. Therefore, the contrast in colour shows the static cone and the mobilised mass.

The Mohr–Coulomb simulations were carried out using the critical state parameters (\(\varphi ^{\prime }_{\mathrm{cs}} = 33^\circ\), \(\psi = 0^\circ\)). The development of the primary failure surface is fast and defines the static cone with small strains and the mobilised mass which takes the form of a sliding wedge. During the collapse, the mobilised mass slides as a rigid body along the primary failure surface and crumbles upon contact with the base. Figure 10a shows the collapse at \(T = 0.3\) s in which the sliding wedge has started to crumble upon contact with the base. Unlike *Column 1*, the primary failure surface is not straight but concave. The wedge slides and rotates along the concave failure surface as a rigid body (\(T = 0.3\) s) but, unlike *Column 1*, a secondary failure surface is observed. However, the size of the primary failure surfaces for *Column 1* and *Column 2*, on which energy is dissipated by frictional contact, are similar. Unlike for *Column 1*, The mobilised mass is also more significant than the static cone. The friction between the mobilised mass and the static regions dissipates energy and slows down the mobilised until its is static. The toe of the wedge rapidly loses momentum due to the frictional contact with the base. The momentum of the mobilised mass is significant due to its mass and flows on (\(T = 0.6\)). During this phase, the flow exhibits multiple smaller flows due to multiple shear zone. The mobilised mass then flows gradually before losing momentum (\(T = 0.8\) s). The stabilisation of the mobilised mass is built bottom-up by successive layers (\(T = 0.8 - 2.5\) s) and is shown in Fig. 10b. The final run-out distance is significantly larger than the experimental prediction (\(r^* = 4.0\)).

The simulations with the Mohr–Coulomb Strain Softening model were carried out for two initial densities (loose and dense) with two different sets of model parameters (loose: \(\varphi ^{\prime }_{\mathrm{peak}} = 39^\circ\), \(\psi _{\mathrm{peak}} = 6^\circ\), dense: \(\varphi ^{\prime }_{\mathrm{peak}} = 50^\circ\), \(\psi _{\mathrm{peak}} = 25^\circ\)). The results for the loose case show a fast developing failure surface. As for the Mohr–Coulomb case with critical state parameters, the failure surface is concave and has a similar shape. Figure 10c shows the collapse at \(T = 0.3\) s in which the sliding wedge has started to crumble upon contact with the base. The primary and secondary failure surfaces have similar shapes as for the Mohr–Coulomb simulation. The flow is then progressively slow down by frictional contact with static layers (\(T = 0.6 -0.8\) s). As previously noticed for the Mohr–Coulomb case, the multiple flow surface appear in the flowing mass. The avalanching process then gradually takes place eroding the summit of the column (\(T = 0.8 - 2.5\) s). The deposition profile (\(T = 2.5\)), as shown in Fig. 10d, has a shape similar to that of the critical state Mohr–Coulomb. The deposition profile is very similar to the Mohr–Coulomb one. The run-out distance is the same (\(r^{*} = 4.0\)) and is much larger than the experimental prediction. The results for the dense case show a fast development of the primary failure surface. However, the failure angle is larger and is planar. Figure 9e shows the results at \(T = 0.3\) s. The newly-formed wedge slides down until the toe reaches the base layer and crumbles (\(T = 0.6\) s). Successive layers of stabilised mass build up from the base to the surface (\(T = 0.8\)), while an avalanching process starts at the summit of the static cone. These two processes continue until the mobilised mass is stabilised (\(T = 2.5\) s). The deposition profile has a run-out distance similar to the ones observed for the other two Mohr–Coulomb cases (\(r^* = 4.0\)). This can be explained by the large inertia of the mobilised mass.

The simulations with Nor-Sand were carried out for two initial densities (loose and dense) but with a unique set of model parameters as Nor-Sand includes the void ratio as a model variable. The results for the loose case show that the development of the primary failure surface is slower than for the Mohr–Coulomb cases, albeit not as significantly slower as for *Column 1*. Figure 10g shows the collapse at \(T = 0.3\) s. Unlike for the Mohr–Coulomb cases, the mobilised mass is subjected to a planar primary failure surface and a multitude of minor secondary ones which divide the mobilised mass into blocks. The blocks then slide while being distorted (\(T = 0.5\) s) and finally crumble upon contact with the base layer (\(T = 0.7\) s). The material then flows on the horizontal surface building the successive static layers (\(T = 0.9\) s). Figure 10f shows the final deposition profile which has an angle of repose close to the critical state angle. The run-out distance (\(r^* = 2.3\)) is shorter than for the Mohr–Coulomb cases and is caused by additional energy dissipation of plastic deformation modelled by Nor-Sand. The run-out distance is within the range of the experimental predictions [21].

The results for the dense case with Nor-Sand show a fast developing and steep failure surface, albeit slower than for the loose case. Figure 10i shows the collapse at \(T = 0.3\) s. The high density of the material causes it to dilate and harden. As the material hardens, the slope of the failure surface increases. Therefore, the mobilised and flowing mass is reduced. Once this reduced mobilised has reached the base and stabilised, an intensive avalanching phase starts (\(T = 0.5\) s) in which the static cone continuously shreds off layers of material (\(T = 0.7 - 0.9\) s). This process is slow and was still continuing at \(T = 2.5\) s as shown in Fig. 10j. The shredding affected mainly the summit of the static cone. The material then free falls or flows on the upper part of the deposited mass. It stabilises on it before reaching the toe of the flow. Therefore, the final run-out distance (\(r^* = 2.2\)) is the one at \(T = 2.5\) s. Unlike for *Column 1*, the run-out distance of the dense case for *Column 2* is comparable to the loose case and within the range of the experimental results [21]. This difference comes from comparable mobilised masses as is explained later in the paper.

## 5 Tracking individual material points

Figure 11a shows the flow paths for the simulations of *Column 1* with the Mohr–Coulomb Strain Softening model and for both densities. The results show that only the two upper material points were mobilised and flowed with a concave path. The material point located in the top corner next to the free surface (MP 774) had the same flow path both in time and in space for both the loose and the dense case. The material point located at mid-distance and a the top of the column (MP 290) had a similar path for the loose and the dense case, but it travelled a shorter distance for the dense case. The two other material point (MP 5633 and 1521) were not mobilised in the collapse.

Figure 11b shows the flow path for simulations of *Column 2* with the Mohr–Coulomb Strain Softening model. The results that three out of the four material points were mobilised for the loose case, whereas only two were for the dense case. Their flow paths are concave as they also were for *Column 1*. The flow paths of the two material points located at the top of the column (MP 6530 and 5520) show the same concave flow paths.

Figure 11c, d show the development of displacements over time of the four material points respectively for *Column 1* and *Column 2*. It shows when the material points were mobilised and immobilised. Two differences appear between the loose and the dense case with a Mohr–Coulomb Strain Softening model. The first difference is whether a material point is mobilised by the primary failure surface or not. It can be seen that MP 290 and MP 5748, respectively for *Column 1* and for *Column 2*, are only mobilised for the loose case. The second difference is whether an avalanching process takes place or not. It can be seen that MP 290 in *Column 1* is mobilised by the primary failure surface for the loose case but by the avalanching process for the dense case. Furthermore, it is mobilised later in case of avalanching.

*Column 1*and 3.5 m/s for

*Column 2*. It can also be seen that the avalanching process took place in

*Column 1*after the stabilisation of initially mobilised mass; material point MP 290 in

*Column 1*was mobilised at

*T*= 0.8 s and was stabilised at

*T*= 2.5 s. Note that a material point which was mobilised by the primary failure in both columns and for both densities exhibited the exact same flow path, displacement and velocity. The only difference between the loose and the dense case is whether a material point is mobilised or not.

Figure 12a shows the flow paths for *Column 2* with Nor-Sand. The results show that two material points (MP 774 and 290) were mobilised for the loose case as for the Mohr–Coulomb Strain Softening simulation, whereas only one of the four material point (MP 290) was mobilised for the dense case. The flow paths with Nor-Sand were more complex than for the Mohr–Coulomb Strain Softening cases and were due to rapid changes in density which were directly taken into account by Nor-Sand. The flow path of the loose sand with Nor-Sand resembled the flow path of the loose sand with Mohr–Coulomb Strain Softening. However, the flow path of the dense case with Nor-Sand is different as it predicts an intensive avalanching process.

Figure 12b shows the flow paths for *Column 2* with Nor-Sand and for both densities. The results show that two out of the four material points (MP 6350 and 5522) were mobilised. Unlike Mohr–Coulomb Strain Softening, Nor-Sand predicted different flow paths for loose and dense cases. The loose case with Nor-Sand resembles the loose case with Mohr–Coulomb Strain Softening, albeit the primary failure surface was steeper.

Figure 12c, d show the development of the displacements over time for both *Column 1* and *Column 2* with Nor-Sand. Differences between the loose and the dense case can be seen for *Column 1* in terms of mobilisation of material points and their stabilisation. It can be seen that the material point located in the top corner (MP 774) travelled further and was immobilised later for the loose case than for the dense case. These differences are limited for *Column 2*.

Figure 12e, f show the development of the velocities for simulations with Nor-Sand. The results show some fluctuation in the evolution of the velocities. This is due to the inclusion of density in the constitutive model. As the material flowed, local variation in density appeared and hence difference mechanical responses were predicted by Nor-Sand. The maximum velocities obtained with Nor-Sand were slightly higher than the those obtained with Mohr–Coulomb Strain Softening—2.75 m/s for *Column 1* and 4 m/s for *Column 2*. Furthermore, a lag time of the onset of the failure is observed between the loose and the dense case of *Column 2*. This difference is not observed for *Column 1*.

## 6 Discussion

*g*gravity, \(h_{\mathrm{mob}}^\mathrm{CG}\) the position of the centre of gravity of the mobilised part, \(h_{\mathrm{tot}}^\mathrm{CG}\) the centre of gravity of the column, \(h_{\mathrm{stat}}^\mathrm{CG}\) the centre of gravity of the static part, \(h_0\) and \(r_0\) respectively, the initial height and radius of the column, \(\varphi ^{\prime }\) the friction angle,

*n*the porosity and \(\rho _s\) the specific gravity of the soil.

*Q*the crushing pressure, \(p^{\prime }\) the pressure, \(R = 1\) a fitting parameter,

*e*the void ratio, \(e_{\mathrm{min}}\) and \(e_{\mathrm{max}}\) respectively, the minimum and maximum void ratio and

*p*the mean effective stress.

The potential energy of the mobilised mass was calculated for a 10 cm wide column for which the height was increased (\(r_0\) = 1 m and \(h_0 = a \times r_0\)). The material properties were taken from Fern et al. [12] and are typical of silica sand (\(\rho _s\) = 2700 kg/m\(^3\), \(\varphi ^{\prime }_{\mathrm{cs}}\) = 33\(^ \circ\), *Q* = 10 MPa, \(e_{\mathrm{min}}\) = 0.500, \(e_{\mathrm{max}}\) = 0.946). The pressure can be estimated by considering the self-weight of the mobilised mass through an iterative process as the volume of the mobilised mass depends on the friction angle. However, the pressure is known to be low and has a limited influence of the potential energy of the mobilised mass and was assumed to be 1 kPa. The consequence is a higher dilatancy angle, albeit limited. Figure 14 shows the potential energy of the mobilised mass for relative densities ranging from 10 to 90 %. The figure shows that the increase in the initial potential energy of the mobilised mass is bi-linear in a log–log plane commonly used in the literature (i.e. [19]). The transition between small and large columns changes with density and occurs in the region of *a* = 1.0 and is consistent with the experimental observations [19, 21, 41]. The figure also shows that the influence of the initial density on the relative amount of potential energy in the system is significant for small aspect ratio but not for large one.

The simulations with Mohr–Coulomb predicted similar run-out distances for loose and dense sands and for both *Column 1* and *Column 2*. In contrast, the simulations with Nor-Sand predicted different run-out distances for loose and dense sand for *Column 1* and similar run-out distances for *Column 2*. This difference in predictions can be explained by considering the energy dissipation mechanism. Mohr–Coulomb predicted a sliding rigid wedge for both loose and dense sand which dissipates energy by frictional contact along the primary failure surface and the base. The rigid wedge exhibit limited distortion and hence dissipated little energy. Furthermore, the hardening phase is modelled as elastic and dissipated no energy. Nor-Sand predicted a sliding soft wedge in which intense shearing took place. Therefore, energy was dissipated along the primary failure surface, the base and inside the wedge. Furthermore, it allows plastic deformation to take place during the hardening phase and thus energy was dissipated. The differences between Mohr–Coulomb and Nor-Sand are largely due to their historical development. The Mohr–Coulomb evolved from Coulomb’s frictional law applied to a shear band on which a block is sliding [8] to a failure criteria [26] and then converted to a constitutive model by including an elastic hardening phase. The Mohr-softening Strain Softening is an adaptation of Mohr–Coulomb to accommodate variation in the model parameters in order to satisfy the critical state theory [31] and mimic the mechanical behaviour of soil. Nor-Sand [14] was developed directly from the critical state theory to model the stress–strain relationship of sand. Its energy dissipation law is based on stress-dilatancy theory and allows energy to be dissipated when distorted and this from the very beginning of the shearing process.

## 7 Conclusion

- 1.
The

*initial geometry*controls the size of the column and the amount of energy available in the system. - 2.
The

*constitutive model*defines the failure surface which splits the column into the static cone and the mobilised mass. The total potential energy of the column is split accordingly. The static cone undergoes very small deformations and most of its potential energy is conserved as such. The mobilised part undergoes very large deformation and most of its potential energy is converted into kinetic energy and dissipated by the constitutive model. Therefore, the influence of the constitutive model is twofold. - 3.
The

*initial density*influences the constitutive models by an enhancement of its mechanical behaviour. It can be captured by an enhancement of the model parameters such as in Mohr–Coulomb or directly through its inclusion as a model variable such as in Nor-Sand. It influences the dilatancy characteristics and consequently the failure angle. The enhancement of the angle of failure by density influences, in turn, the volume of the mobilised mass, its potential energy and, in some cases, the dissipation of that energy. The analysis of the initial potential energy showed that the influence of density is more significant for small columns than for larger ones. It also explains the existence of two families of collapse regimes as suggested by Lajeunesse et al. [18, 19] and Lube et al. [21, 22, 23].

## Notes

### Acknowledgments

This project has received funding from the European Unions Seventh Framework Programme for research, technological development and demonstration under Grant Agreement No. PIAP-GA-2012-324522 and the Swiss National Science Foundation under Grant Agreement P1SKP2 158621.

## References

- 1.Andersen S (2009) Material-point analysis of large-strain problems : modelling of landslides. Ph.D. thesis, Aalborg UniversityGoogle Scholar
- 2.Balmforth NJ, Kerswell RR (2005) Granular collapse in two dimensions. J Fluid Mech 538:399–428. doi: 10.1017/S0022112005005537 MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Bandara S (2013) Material point method to simulate large deformation problems in fluid-saturated granular medium. Ph.D. thesis, University of CambridgeGoogle Scholar
- 4.Bardenhagen SG, Kober EM (2004) The generalized interpolation material point method. Comput Model Eng Sci 5(6):477–495Google Scholar
- 5.Been K, Jefferies M (1985) State parameter for sands. Géotechnique 35(2):99–112CrossRefGoogle Scholar
- 6.Bolton M (1986) The strength and dilatancy of sands. Géotechnique 36(1):65–78CrossRefGoogle Scholar
- 7.Chen W, Qiu T (2012) Numerical simulations for large deformation of granular materials using smoothed particle hydrodynamics method. Int J Geomech 12(April):127–135. doi: 10.1061/(ASCE)GM.1943-5622.0000149 CrossRefGoogle Scholar
- 8.Coulomb C (1776) Essai sur une application des règles de maximis & minimis à quelques problèmes de statique relatif à l’architecture. De l’Imprimerie RoyaleGoogle Scholar
- 9.Crosta GB, Imposimato S, Roddeman D (2009) Numerical modeling of 2-D granular step collapse on erodible and nonerodible surface. J Geophys Res Solid Earth 114:1–19. doi: 10.1029/2008JF001186 CrossRefGoogle Scholar
- 10.Daerr A, Douady S (1999) Sensitivity of granular surface flows to preparation. Europhys Lett 47(August):324–330CrossRefGoogle Scholar
- 11.Darve F, Servant G, Laouafa F, Khoa H (2004) Failure in geomaterials: continuous and discrete analyses. Comput Methods Appl Mech Eng 193(27–29):3057–3085. doi: 10.1016/j.cma.2003.11.011 CrossRefzbMATHGoogle Scholar
- 12.Fern J, Sakanoue T, Soga K (2015) Modelling the shear strength and dilatancy of dry sand in triaxial compression tests. In: Soga K, Kumar K, Biscontin G, Kuo M (eds) Geomechanics from micro to macro. Taylor & Francis, London, pp 673–678Google Scholar
- 13.Girolami L, Hergault V, Vinay G, Wachs A (2012) A three-dimensional discrete-grain model for the simulation of dam-break rectangular collapses: comparison between numerical results and experiments. Granul Matter 14:381–392. doi: 10.1007/s10035-012-0342-3 CrossRefGoogle Scholar
- 14.Jefferies M (1993) Nor-Sand: a simple critical state model for sand. Géotechnique 43(1):91–103MathSciNetCrossRefGoogle Scholar
- 15.Jefferies M, Been K (2006) Soil liquefaction: a critical state approach. Taylor & Francis, LondonCrossRefGoogle Scholar
- 16.Kumar K (2014) Multi-scale multiphase modelling of granular flows. Ph.D. thesis, University of CambridgeGoogle Scholar
- 17.Lacaze L, Phillips JC, Kerswell RR (2008) Planar collapse of a granular column: experiments and discrete element simulations. Phys Fluids 20(6):063,302. doi: 10.1063/1.2929375 CrossRefzbMATHGoogle Scholar
- 18.Lajeunesse E, Mangeney-Castelnau A, Vilotte JP (2004) Spreading of a granular mass on a horizontal plane. Phys Fluids 16(7):2371–2381. doi: 10.1063/1.1736611 CrossRefzbMATHGoogle Scholar
- 19.Lajeunesse E, Monnier JB, Homsy GM (2005) Granular slumping on a horizontal surface. Phys Fluids 17(10):103,302. doi: 10.1063/1.2087687 CrossRefzbMATHGoogle Scholar
- 20.Liang D, He XZ (2014) A comparison of conventional and shear-rate dependent Mohr–Coulomb models for simulating landslides. J Mt Sci 11(6):1478–1490. doi: 10.1007/s11629-014-3041-1 MathSciNetCrossRefGoogle Scholar
- 21.Lube G, Huppert H, Sparks R, Freundt A (2005) Collapses of two-dimensional granular columns. Phys Rev E 72(4):1–10. doi: 10.1103/PhysRevE.72.041301 CrossRefGoogle Scholar
- 22.Lube G, Huppert H, Sparks R, Hallworth M (2004) Axisymmetric collapses of granular columns. J Fluid Mech 508:175–199. doi: 10.1017/S0022112004009036 CrossRefzbMATHGoogle Scholar
- 23.Lube G, Huppert H, Sparks RSJ, Freundt A (2007) Static and flowing regions in granular collapses down channels. Phys Fluids 19(4):043301. doi: 10.1063/1.2712431 CrossRefzbMATHGoogle Scholar
- 24.Mast C, Arduino P, Mackenzie-Helnwein P, Miller GR (2014) Simulating granular column collapse using the material point method. Acta Geotech. doi: 10.1007/s11440-014-0309-0 Google Scholar
- 25.Mitchell J, Soga K (2005) Fundamentals of soil behavior, 3rd edn. Wiley, HobokenGoogle Scholar
- 26.Mohr CO (1928) Welche Umstaad Bedingen des Elastizitaetsgrenzen und den Bruch eines material? Abhandlingen aus dem Gebiete der Technischen Mechanik, 3rd edn. Ernst und Sohn, BerlinGoogle Scholar
- 27.Nova R (1982) A constitutive model for soil under monotonic and cyclic loading. In: Pande GN, Zienkiewicz C (eds) Soil mechanics—transient and cyclic loading. Wiley, Chichester, pp 343–373Google Scholar
- 28.Potts D, Zdravkovic L (2001) Finite element analysis in geotechnical engineering—application. Thomas Telford Ltd, LondonCrossRefGoogle Scholar
- 29.Pouliquen O (1999) Scaling laws in granular flows down rough inclined planes. Phys Fluids 11(3):542–548MathSciNetCrossRefzbMATHGoogle Scholar
- 30.Roscoe KH, Schofield A (1963) Mechanical behaviour of an idealised wet clay. In: 2nd European conference on soil mechanics and foundation engineering. Wiesbaden, pp 47–54Google Scholar
- 31.Roscoe KH, Schofield A, Wroth CP (1958) On the yielding of soils. Géotechnique 8(1):22–53. doi: 10.1680/geot.1958.8.1.22 CrossRefGoogle Scholar
- 32.Rowe PW (1962) The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. In: Proceedings of the Royal Society A: mathematical, physical and engineering sciences, vol 269. The Royal Society, pp 500–527Google Scholar
- 33.Schofield A, Wroth P (1968) Critical state soil mechanics, 2nd edn. McGraw-Hill, LondonGoogle Scholar
- 34.Solowski WT, Sloan SW (2013) Modelling of sand column collapse with material point method. In: Pande GN, Pietruszczak S (eds) Computational geomechanics 2013, vol 553, pp 698–705. http://ogma.newcastle.edu.au/vital/access/manager/Repository/uon:17186
- 35.Solowski WT, Sloan SW (2015) Evaluation of material point method for use in geotechnics. Int J Numer Anal Meth Geomech 39(7):685–701. doi: 10.1002/nag.2321 CrossRefGoogle Scholar
- 36.Staron L, Hinch EJ (2005) Study of the collapse of granular columns using two-dimensional discrete-grain simulation. J Fluid Mech 545(–1):1. doi: 10.1017/S0022112005006415 CrossRefzbMATHGoogle Scholar
- 37.Sulsky DL, Chen Z, Schreyer H (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196MathSciNetCrossRefzbMATHGoogle Scholar
- 38.Sulsky DL, Zhou SJ, Schreyer H (1995) Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 87:236–252CrossRefzbMATHGoogle Scholar
- 39.Taylor D (1948) Fundamentals of soil mechanics. Wiley, New YorkGoogle Scholar
- 40.Utili S, Zhao T, Houlsby G (2015) 3D DEM investigation of granular column collapse: evaluation of debris motion and its destructive power. Eng Geol 186:3–16CrossRefGoogle Scholar
- 41.Warnett JM (2014) Stationary and rotational axisymmetric granular column collapse. Jason Warnett School of Engineering. Ph.D. thesis, University of WarwickGoogle Scholar
- 42.Wieckowski Z (2004) The material point method in large strain engineering problems. Comput Methods Appl Mech Eng 193(39–41):4417–4438. doi: 10.1016/j.cma.2004.01.035 CrossRefzbMATHGoogle Scholar
- 43.Yerro A, Alonso E, Pinyol N (2015) The material point method for unsaturated soils. Géotechnique 65(3):201–217. doi: 10.1680/geot.14.P.163 CrossRefGoogle Scholar
- 44.Zenit R (2005) Computer simulations of the collapse of a granular column. Phys Fluids 17(3):031,703. doi: 10.1063/1.1862240 CrossRefzbMATHGoogle Scholar

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