# Unified modelling of granular media with Smoothed Particle Hydrodynamics

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## Abstract

In this paper, we present a unified numerical framework for granular modelling. A constitutive model capable of describing both quasi-static and dynamic behaviours of granular material is developed. Two types of particle interactions controlling the mechanical responses, frictional contact and collision, are considered by a hypoplastic model and a Bagnold-type rheology relation, respectively. The model makes no use of concepts like yield stress or flow initiation criterion. A smooth transition between the solid-like and fluid-like behaviour is achieved. The Smoothed Particle Hydrodynamics method is employed as the unified numerical tool for both solid and fluid regimes. The numerical model is validated by simulating element tests under both quasi-static and flowing conditions. We further proceed to study three boundary value problems, i.e. collapse of a granular pile on a flat plane, and granular flows on an inclined plane and in a rotating drum.

## Keywords

Bagnold rheology Granular material Hypoplasticity Rotating drum Smoothed Particle Hydrodynamics Solid/fluid transition## 1 Introduction

Granular materials are widely involved in various industry processes and natural phenomena. In industry, one is often interested in the granular flow capacity through complex geometry without blockage [32], as well as the forces exerting on structures [13]. In the field of geophysics, landslides and debris flows are natural hazards possessing great threat to human society. It is therefore important to predict the flowing mass, velocity, run-out distance and impact force of these hazards on structures. An appropriate constitutive model and an adequate numerical method capable of providing high-quality simulation of granular media are of great interest.

As granular materials are collections of rigid macroscopic particles, the discrete element method (DEM), based on elementary mechanical principals idealized from particle interactions [14, 47], is considered suitable for granular modelling. However, the DEM traces the motion and mechanical response of each particle, which makes the computational cost still prohibitive in real-scale industrial and geophysical simulations. On the other hand, in traditional continuum approaches, granular materials are considered as continuum media and can be described using constitutive models and field variables. Therefore, the reliability of continuum approaches largely depends on how exact the constitutive models are in portraying the granular behaviours.

It is well known that, depending on the geometry and loading condition, a granular body can behave like a solid or liquid [22, 15]. Many works have been carried out to study the granular behaviours in the two distinct regimes. Theories of soil mechanics, e.g. elastoplasticity and hypoplasticity, capture the salient behaviours of granular materials in the quasi-static solid-like regime. Constitutive models for the quasi-static regime are usually rate-independent, making them inappropriate for the modelling of granular flows. Rate-dependent viscoplastic models consider viscous strain in the post-failure regime and usually describe a progressive evolution from plastic flow to viscous flow [39]. However, previous studies show that a granular flow can happen without satisfying the failure condition [39, 16, 25] and sometimes is characterized by an abrupt solid/fluid transition [18, 2]. On the other hand, models using a viscous relation based on fluid dynamics are widely used to simulate the flow regime. For instance, non-Newtonian rheology finds its application in granular flow modelling [20, 40]. Another example of granular rheology, more elaborated and sophisticated, is the \(\mu (I)\) model based on dimension analysis and experimental observations [2, 24]. The \(\mu (I)\) model draws much attention recently in granular flow simulations [11, 46, 5]. Usually, in numerical simulations, a granular rheology is used together with a yield stress. Around the yield stress, the viscosity changes drastically from a low physical value to an artificial numerical one. The issue for granular rheology is that the quasi-static solid material behaviours inside the yield surface are undefined.

In industrial and geophysical problems, the granular flows often consist of flowing and stagnant regions. In these two regions, the material can be described with the models mentioned above from soil mechanics and fluid dynamics, respectively. A straightforward approach is to model the two regimes with different material models. This method requires complex numerical techniques making use of different solvers for solid and fluid. Identification of the solid/fluid interfaces is also necessary. Moreover, the solid/fluid transition needs to be explicitly given to capture the evolution of solid/fluid interfaces. Therefore, a more rational way is to unify the solid-like and fluid-like responses as well as the solid/fluid transition in one single constitutive model. A few efforts have been proposed [1, 17, 9], by including rate-dependency in conventional elastoplastic models. In these models, flows can happen only after yielding. One noteworthy model is presented in [39, 38], which is a combination of an elastoplastic model and a Bingham rheology. In this model, viscous flows can be activated before the plastic limit criterion being reached by checking the second-work instability [16]. Nevertheless, the development of well-established unified relations describing both solid-like and fluid-like behaviours of granular materials is still an ongoing undertaking.

Granular flow modelling also has some special requirements for numerical methods. An eligible method should be capable of simulating problems of solid mechanics and fluid dynamics at the same time with an acceptable complexity. Both large deformation of solids and free surface flow of fluids need to be modelled. Conventional grid-based numerical methods such as finite element method (FEM) and finite differential method (FDM) have difficulty in taking such task. Recently developed particle-based methods, e.g. Smoothed Particle Hydrodynamics (SPH), material point method (MPM) and particle FEM (PFEM), are more suitable for granular modelling. Nevertheless, previous studies apply either rate-independent models for quasi-static state [7, 36, 30], or pure rheology models for flow regime [11]. Unified numerical modelling of granular media still calls for further investigation.

In this paper, we present a unified modelling of granular media using the SPH method. A rational yet simple constitutive relation capable of describing solid-like, fluid-like and solid/fluid transition behaviours of granular materials is adopted. The constitutive model makes use of a hypoplastic model and an extended 3-D Bagnold-type relation. It captures the failure and critical state properties of granular materials in the quasi-static regime, as well as the viscous behaviour and collision-induced dilatation in dynamic granular flows. The meshfree SPH method is employed as it proves applicable in both large deformation analysis and fluid dynamics. SPH kernel gradient renormalization is applied to treat the particle inconsistency, thus improve the accuracy. We first validate the unified method with element tests. Some granular flow problems with common configuration are then numerically simulated.

## 2 A unified constitutive model for granular materials

### 2.1 Constitutive model framework

### 2.2 Quasi-static solid-like response

Hypoplastic constitutive models are based on nonlinear tensorial functions with the major advantage of simple formulation and few parameters. In this paper. we will embark on the first hypoplastic model with critical state proposed by Wu et al. [51]. Recently, some improved hypoplastic models have been available [27, 45], which aim to improve the dependence of stiffness on pressure and density. However, such models are rather complex in formulation with more model parameters. Since the initiation of granular flow is mainly dictated by failure rather than deformation, the original model by Wu et al. with its simple formulation and few parameters offers a better choice for the unified modelling of granular media.

*a*is a material constant and \(D_c\) is the modified relative density

*e*, \(e_{{\mathrm {min}}}\) and \(e_{{\mathrm {crt}}}\) are the initial, minimum and critical void ratio, respectively. Based on laboratory tests, the critical void ratio \(e_{{\mathrm {crt}}}\) and material constant

*a*can be related to the stress level [51]

### 2.3 Extended 3-D Bagnold-type rheology

Many rheology models have been used in granular flow modelling, from simple Bingham model [39, 38, 26] to more sophisticated ones [24, 46, 1]. These models consider the combined frictional and collisional interactions from a phenomenological point of view by a single viscosity coefficient. In our model, the dynamic part only accounts for the collision interaction; therefore, the viscous relations mentioned above are inappropriate.

*x*,

*z*are Cartesian coordinates. Moreover, a collision-induced pressure \(p_{\mathrm{v}}\), termed dispersive pressure, is found proportional to the shear stress in rapid shearing [4, 3]

*C*is the solid volume fraction. Following the general form, we propose an extended Bagnold-type rheology for granular flows

### 2.4 The unified model for granular materials

In general granular problems where particle interactions possibly include friction and collision, both the stress parts take effect simultaneously. The proportions of frictional and collisional stresses depend on the kinematics, loading and boundary conditions of flows. For instance, from very slow motion to rapid flow, the dynamic part becomes more and more significant. This results in an increase in the dispersive pressure \(p_v\) and a tendency of dilation, thus leads to a relaxation in the normal stress of frictional contact. Therefore, the hypoplastic stress part tends to decrease with the increase of shear rate. This dynamic balance can be achieved in numerical simulations.

The presented model bridges the granular solid-like and fluid-like behaviours and achieves a natural solid/fluid transition without any flow initiation criterion. The frictional and collisional stress parts evolve concurrently depending on the flow state, the loading and boundary conditions. Complex concepts such as yield stress and strain decomposition are not required in the model. Therefore, the numerical implementation is simplified. The unified model has 13 material parameters: \(c_1\sim c_4\) related to hypoplasticity, \(e_{{\mathrm {min}}}\), \(q_1\sim q_3\) and \(p_1\sim p_3\) for critical state, and two parameters \(\alpha _i\) and \(k_{\mathrm{v}}\) for the dynamic part. According to [4], the parameter \(k_{\mathrm{v}}\) can be further calculated from particle density, particle diameter and solid volume fraction.

## 3 SPH modelling of granular material

Granular flows often involve large deformation from solid mechanics point of view and free surface flow from a fluid dynamics perspective, which are cumbersome to handle with conventional numerical methods. The meshfree and Lagrangian properties of SPH make it an appealing method for granular flow modelling [11, 7, 36, 37, 12].

### 3.1 The governing equations

### 3.2 The SPH formulations

The SPH is a Lagrangian meshfree method with no computational grid. The whole computational domain \(\Omega\) is discretized with a set of particles, which carry the physical properties and computational variables, and move with material velocities. By tracking the movement of the particles and the evolution of the carried variables, the considered problem can be solved numerically. Because the SPH applies an updated Lagrangian formulation, large deformation problems in solid mechanics as well as free surface tracking in fluid dynamics are treated naturally. It is therefore an excellent choice for the numerical modelling of granular media, where the material may come across large deformation in quasi-static motion, and rapid free surface flow after failure.

*h*is a smoothing length used to control the support size of

*W*. The kernel function used through this paper is a Wendland C\(^6\) function [19, 49] for which the compact support has a radius of 2

*h*. The field function’s spatial derivatives are obtained by substituting \(\nabla f({\varvec{x}})\) into Eq. (16)

*i*has an associated kernel function \(W_i\) centred at \({\varvec{x}}_i\). Replacing the continuous integrations in Eqs. (16) and (17) with particle summation, the discrete approximations of the field function and its derivatives are written as

*n*is the number of particles within the support domain of particle

*i*and \(m_j/\rho _j\) denotes the volume of material represented by particle

*j*. Through SPH simulations, the mass of a particle is usually constant but its density can evolve according to the varying inter-particle spacing, resulting to constantly changing particle volume. \(\nabla _iW_{ij}\) is the gradient of kernel function \(W_{ij}\) evaluated at the location \({\varvec{x}}_i\).

### 3.3 The correction of kernel gradient

The continuous SPH kernel interpolation in Eq. (16) theoretically ensures second-order accuracy for interior regions. That is, constant and linear functions can be reproduced exactly. However, this \(C^0\) and \(C^1\) consistency are not always satisfied in the SPH particle approximation, especially when particle distributions are irregular, or the particle support domain is truncated by boundaries [28]. This deficiency, termed particle inconsistency, is the direct cause of the relatively low accuracy and slow convergence rate in the original SPH method. Usually the lack of \(C^1\) consistency is more hazardous, because the discrete forms of the governing Eqs. (20) and (21) all make use of the kernel gradient \(\nabla W\). Many corrections have been proposed to improve the accuracy of kernel gradient approximation. In the present study, we use the renormalization technique [6, 34] to enforce the \(C^1\) consistency, which is briefly given as follows.

*i*

*i*. \({\varvec{L}}\) is the renormalization matrix in the following form

### 3.4 Numerical implementation and boundary conditions

*c*is the artificial speed of sound used to control the size of the time step, and \(\chi _{\mathrm{CFL}}\) is the Courant condition coefficient. In this paper,

*c*and \(\chi _{\mathrm{CFL}}\) are taken as 80 m/s and 0.05, respectively.

In the unified model, the hypoplastic stress part is history-dependent. Therefore, the hypoplastic stress tensor \({\varvec{\sigma }}_{\mathrm{h}}\) is calculated by integrating hypoplastic stress rate \(\dot{{\varvec{\sigma }}}_{\mathrm{h}}\) using the same Predictor–Corrector scheme. As long as the time step is small enough, the direct stress integration of the hypoplastic stress is accurate sufficiently [36]. Since there is no explicit failure in hypoplasticity, complex stress integration and return-mapping algorithms in elastoplasticity are unnecessary.

Two kinds of boundary conditions are considered in this paper, i.e. periodic boundary and non-slip solid boundary. The periodic boundary condition in SPH is straightforward, as shown in [11, 21]. The treatment of solid boundary condition is still a challenge in SPH computations. In this paper, we employ a boundary particle method developed for geomechanical applications [35]. In this method, the solid boundary is discretized with boundary particles, which take part in the SPH approximation like real particles but keep fixed or move with prescribed motions. The velocity and stress tensor at boundary particles are extrapolated from the real granular particles. We find that this boundary treatment method works properly in our simulations.

## 4 SPH element tests

In this section, the proposed unified constitutive model and the SPH implementation are validated using two element tests. For simplicity, velocity restrictions are imposed on SPH particles to reproduce a uniform deformation condition in the test sample, analogous to an element test in the FEM.

### 4.1 Quasi-static undrained simple shear test

Constitutive parameters used in the simple shear test

\(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(e_{\min }\) | \(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) | \(k_{\mathrm{v}}\) | \(\alpha _i\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(–) | (–) | (–) | (–) | (–) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (–) | (\(^\circ\)) |

\(-\)50.0 | \(-\)629.6 | \(-\)629.6 | 1220.8 | 0.597 | 0.53 | 0.45 | \(-\)0.0018 | 1.0 | \(-\)0.4 | \(-\)0.0001 | 0.01 | 35.0 |

### 4.2 Granular sheared in an annular shear cell

Constitutive parameters used in the annular shear test

\(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(e_{\min }\) | \(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) | \(\alpha _i\) |
---|---|---|---|---|---|---|---|---|---|---|---|

(–) | (–) | (–) | (–) | (–) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (\(^\circ\)) |

\(-\)50.0 | \(-\)746.6 | \(-\)746.6 | 1855.1 | 0.563 | 0.65 | 0.55 | \(-\)0.11 | 1.0 | \(-\)0.24 | \(-\)0.013 | 27.0 |

List of solid volume fraction *C*, void ratio *e*, residual stresses and viscous coefficient \(k_{\mathrm{v}}\) in the four simulations

Solid volume fraction | 0.461 | 0.483 | 0.504 | 0.524 |

Void ratio | 1.169 | 1.070 | 0.984 | 0.908 |

\(\sigma _{xz}\) (Pa) | 0.0 | 0.0 | 0.0 | 36.5 |

\(\sigma _{zz}\) (Pa) | 0.0 | 0.0 | 0.0 | 83.2 |

\(k_{\mathrm{v}}\) | 0.0018 | 0.0027 | 0.0041 | 0.0065 |

In the numerical simulations a velocity field of \(v_x=2\kappa z\) and \(v_z=0\) m/s is imposed, where \(\kappa\) is the desired shear rate. In this work, the range of shear rate \(\kappa\) is 10 \(\sim\) 500. At the beginning, the model is given a shear rate of \(\kappa =10\) for 0.5 s, to make sure that the granular flow fully develops and the critical state is reached. Then \(\kappa\) is linearly increased to 500 in 20 s. Due to the extremely large shear strain, any stress part from the hypoplastic model is the residual stress after critical state being reached.

Table 3 lists the residual stress from simulations with different solid volume fraction after the first 0.5 s of shearing. It can be seen that for \(C=0.461\), \(C=0.483\) and \(C=0.504\), the residual stresses are zero, corresponding to a full liquefaction (or gasification since the interstitial fluid is air). It means that the granular particles lose frictional contact from a constitutive modelling point of view. For the dense case \(C=0.524\), a nonzero residual stress is obtained regardless of the magnitude of shear strain. According to [4], viscous coefficient \(k_{\mathrm{v}}\) can be calculated from particle density \(\rho _s\), particle diameter \(d_s\) and volume fraction *C*. The obtained viscous coefficient \(k_{\mathrm{v}}\) for the four simulations is also listed in Table 3. Since \(\rho _s\) and \(d_s\) are constants for a granular material, the solid volume fraction (void ratio) is significant to the granular flowing property as it has great influence on \(k_{\mathrm{v}}\).

## 5 Application to boundary value problems

Two element tests are performed in Sect. 4, where the velocity fields are prescribed to reproduce uniform deformation conditions. The tests give preliminary validation of our unified approach. In this section, the proposed unified modelling method is applied to real granular flow problems.

### 5.1 Granular flow on an incline

*H*. In this section, we study characteristics of the granular flow on different inclinations.

#### 5.1.1 Analysis based on the unified model

*z*should be hydrostatic

*z*direction requires \(\sigma _{zz}=\sigma _{zz}^h=\rho g(H-z)\cos \theta\). As a result, the condition for static state is \(\sigma _{xz}=\sigma _{xz}^h\ge \rho g(H-z)\sin \theta\), which means \(\theta \le \phi\).

One important assumption in the above analysis is \(\sigma _{xz}^h/\sigma _{zz}^h=\tan \phi\) at failure and subsequent flow state. When describing granular flows in the framework of depth-averaged equations, Savage and Hutter employ the same assumption [42]. However, some discrete simulations suggest that \(\sigma _{xz}^h/\sigma _{zz}^h=\sin \phi\) [2, 56], indicating a different failure slip line. Besides, the actual failure surface of the hypoplastic model is slightly different from that of Mohr–Coulomb model, which may result to minor violation of the frictional law. Nevertheless, the above analysis process is valid in principal, though the exact ratio between \(\sigma _{xz}^h\) and \(\sigma _{zz}^h\) may be put to further discussion. Considering separately the friction and collision in granular flow gives rise to interesting findings.

#### 5.1.2 Numerical modelling

Material parameters for the granular material on a inclined plate

\(c_1\) | \(c_2\) | \(c_3\) | \(c_4\) | \(e_{\min }\) | \(p_1\) | \(p_2\) | \(p_3\) | \(q_1\) | \(q_2\) | \(q_3\) | \(k_{\mathrm{v}}\) | \(\alpha _i\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(–) | (–) | (–) | (–) | (–) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (–) | (–) | (\(\hbox {kPa}^{-1}\)) | (–) | (\(^\circ\)) |

–66.7 | –832.9 | –832.9 | 1594.5 | 0.597 | 0.53 | 0.45 | –0.0011 | 1.0 | –0.4 | –0.0001 | 0.1 | 33.0 |

### 5.2 Collapse of a granular pile

The collapse of a granular pile on a flat surface is studied in this section. This problem has well-defined initial and boundary conditions and has been studied experimentally and numerically [7, 36, 29]. Previously numerical simulations often apply rate-independent constitutive models, thus do not consider the dynamic effect in granular flows.

### 5.3 Granular flow in a rotating drum

*z*are studied. \(z_y\) is the height where flow state changes and \(z_s\) is the height of the flow surface; \(h=z_s-z_y\) is the flow depth. In the range of \(z<z_y\) the granular mass behaves solid-like, so the velocity profile is linear. The steady velocity profiles in the dynamic flowing layer are given in Fig. 19. On a middle cross section, the velocity direction pointing down the slope is defined as positive. The velocity profiles for different rotating speeds are similar in shape but with different magnitudes. In experiments [31], similar velocity profiles are observed. The simulated flow rates, flow depths and average inclinations are summarized in Table 5. In our flow configuration, the flow rate is calculated by \(Q=0.5\omega\,[R^2-(R-H_0)^2]\), where \(R=D/2\) is the drum radius. It is found that the flow depth

*h*and average inclination \(\theta\) changes almost linearly with respect to the flow rate

*Q*. This observation is also well collaborated with the experimental results [31].

Summary of the numerical modelling of rotating drum

Rotating speed \(\omega\) (s\(^{-1}\)) | 0.5 | 1.0 | 2.0 |

Flow rate | 0.0052 | 0.0104 | 0.0208 |

Flowing depth | 0.046 | 0.054 | 0.065 |

Average inclination \(\theta\) (\(^\circ\)) | 21.48 | 22.43 | 24.81 |

## 6 Conclusions

This paper presents a unified framework for modelling granular media. The hypoplastic model and Bagnold-type relation, respectively, employed to consider the granular frictional contact and collision, are combined to obtain a complete constitutive model for the entire deformation and flow regime of granular media. The unified model makes no use of explicit determination of solid/flow state or flow initiation criterion. It covers the whole spectrum of granular state from quasi-static motion to rapid granular flow. This conversion of motion state is achieved by the coupled evolution of the frictional contact and collision stress parts. The solid-like and fluid-like behaviours of granular materials, as well as the transition between the solid-like and fluid-like states, are well described by the unified constitutive model.

The SPH is proved ideal for modelling both solid-like and fluid-like behaviours within a consistent numerical scheme. The numerical simulation of two element tests shows that the unified approach captures the salient feature of the quasi-static and flowing states of granular materials. We proceed to study three boundary value problems: two granular flows, one on an incline the other in a rotating drum, and a granular pile collapse problem. The following observations are made: (1) In the case of granular flow down an inclined plane, we have obtained a range of inclinations during which a steady dense granular flow is observed. The numerical simulation agrees well with the analytical solution derived from the unified model. Moreover, the solutions to this problem in the present model are well collaborated with those from \(\mu (I)\) model [24, 11]. (2) For the granular pile collapse and the granular flow in rotating drum, the numerical results show wealth of various behaviours, i.e. quasi-static motion, shear band, flow initiation, fully developed granular flow and granular deposition.

The numerical results in this paper are promising to handle the complex behaviour of granular flow in a consistent numerical model. We will apply the unified approach to bulk solids handling in industry and debris flow in nature. However, some aspects, such as particle segregation and hydro-mechanical coupling, still need further investigations to be considered in the unified framework.

## Notes

### Acknowledgments

Open access funding provided by University of Natural Resources and Life Sciences Vienna (BOKU). The authors acknowledge the funding from the European Commission under its People Programme (Marie Curie Actions) of the Seventh Framework Programme FP7/2007-2013/ under Research Executive Agency Grant agreement No. 289911 under title: Multiscale Modelling of Landslides and Debris Flows. The first author wishes to acknowledge the financial support from the Otto Pregl Foundation for Fundamental Geotechnical Research in Vienna.

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