Examining the smooth and nonsmooth discrete element approaches to granular matter

The smooth and nonsmooth approaches to the discrete element method (DEM) are examined from a computational perspective. The main difference can be understood as using explicit versus implicit time integration. A formula is obtained for estimating the computational effort depending on error tolerance, system geometric shape and size, and on the dynamic state. For the nonsmooth DEM (NDEM), a regularized version mapping to the Hertz contact law is presented. This method has the conventional nonsmooth and smooth DEM as special cases depending on size of time step and value of regularization. The use of the projected Gauss‐Seidel solver for NDEM simulation is studied on a range of test systems. The following characteristics are found. First, the smooth DEM is computationally more efficient for soft materials, wide and tall systems, and with increasing flow rate. Secondly, the NDEM is more beneficial for stiff materials, shallow systems, static or slow flow, and with increasing error tolerance. Furthermore, it is found that just as pressure saturates with depth in a granular column, due to force arching, also the required number of iterations saturates and become independent of system size. This effect make the projected Gauss‐Seidel solver scale much better than previously thought. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical simulation of granular materials by an improved discrete element method

In this paper, we present an improved discrete element method based on the non‐smooth contact dynamics and the bi‐potential concept. The energy dissipated during the collisions is taken into account by means of restitution coefficients. The interaction between particles is modelled by Coulomb unilateral contact law with dry friction which is typically non‐associated: during the contact, the sliding vector is not normal to the friction cone. The main feature of our algorithm is to overcome this difficulty by means of the bi‐potential theory. It leads to an easy implement predictor–corrector scheme involving just an orthogonal projection onto the friction cone. Moreover the convergence test is based on an error estimator in constitutive law using the corner stone inequality of the bipotential. Then we present numerical simulations which show the robustness of our algorithm and the various possibilities of the software ‘MULTICOR’ developed with this approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Future continuum models for granular materials in penetration analyses

This paper presents an investigation on future continuum models for granular materials in penetration analyses. A two-dimensional discrete element method has been used to numerically simulate penetration tests on a granular ground. The stress paths of soil elements in the ground have been studied, and then used to highlight the main features of granular materials based on most-advanced knowledge in soil mechanics. The study shows that the penetration makes the soil near the penetrometer undergo a significant changes of stresses in both magnitude and direction. The soil of large deformation rate may arrive at a stress state slightly over the strength envelope obtained from conventional tests. As a result, shear dilatancy, rate-dependency, non-coaxiality and particle crushing are the four main features that future continuum models should capture for granular materials in penetration analyses.     

Mechanism of avalanche precursors in inclining granular layers using a continuum model obtained by discrete element method

How do large shear stress and stress ratio peaks, which are avalanche precursors, occur in granular layers in an inclining box and why do the precursors have periodicities? The answers are our main objectives. The large and small shear stress peaks which are constituted by the large and small sticks and slips are found in our simulation results. Small sticks and slips, which occur after the large slip, change the granular structure until the stress ratio becomes equal to or larger than the frictional coefficient of granular materials. Here the next large slip with the large stick and slip, which shows the precursor, occurs. The precursors which are constituted by the series of the large stick–slip events have naturally periodicities because the stick–slip events are oscillation phenomena. Our 3-dimensional simulation results using the smoothed particle hydrodynamics (SPH) method based on our constitutive equations obtained by the 3-dimensional discrete element method (DEM) present the precursors in the bulk and on the surface and show the difference between them. The precursors are also found in the longitudinal and the depth directions of the granular layer. The simulated periodicities agree with the published experimental results.     

Failure mechanisms in granular media: a discrete element analysis

This paper attempts to numerically validate the concept of diffuse failure using a discrete element method. First, the theoretical background is reviewed, and it is shown how the kinetic energy of a system, initially at rest after a loading history, is likely to abruptly increase under the effect of disturbances. The vanishing of the second-order work thus constitutes a basic ingredient, related to both the pioneering work of Hill (J Mech Phys Solids (6):236–249, 1958) and the notion of bifurcation applied to geomechanics (Vardoulakis and Sulem in Bifurcation analysis in geomechanics, Chapman & Hall Publisher, London, 1995). Discrete numerical simulations were performed on homogeneous three-dimensional specimens, and the three basic conditions that must be satisfied in order to observe a failure mechanism are numerically checked. Finally, this work illustrates the phenomena that are likely to affect in situ slopes, for instance, when the loading (due to weather conditions or human activities) meets the three basic conditions for a failure mechanism to develop.     

Classical and non-classical kinematic fields of two-dimensional penetration tests on granular ground by discrete element method analyses

The penetration test is a widely used in-situ test in geotechnical engineering which mechanism is very important to geomechanics. This paper presents a numerical study on both classic and non-classic kinematic fields in penetration tests on granular ground. A two-dimensional Discrete Element Method (DEM) has been used to simulate penetration tests on a full-size granular ground that is under an amplified gravity and under a K ₀ lateral stress boundary. In addition to classical kinematic variables, i.e. displacement and velocity, a non-classical kinematic variable called the average pure rotation rate (APR), which represents particle sizes and particle rotations (M. J. Jiang et al. Kinematic models for non-coaxial granular materials: Part I: theories. Int J Numer Anal Methods Geomech 2005; 29(7): 643–661), is investigated in the penetration test. The DEM numerical results show that the penetration leads to significant changes in displacement, velocity and APR fields, making the soil near the penetrometer move in complex displacement and APR paths. In comparison to velocity field, APR field is very ‘localized’ in the area close to the penetrometer shoulder during penetration. Based on the normalized tip resistance, the penetration process can be described by three phases of penetration, in which the granular ground undergoes three types of failure mechanism, respectively.     

A discrete element analysis of elastic properties of granular materials

The elastic properties of a regular packing of spheres with different tolerances were evaluated using the discrete element method to elucidate the mechanisms behind the discrepancies between laboratory experiments and theoretical predictions of the classic Hertz-Mindlin contact law. The simulations indicate that the elastic modulus of the packing is highly dependent on the coordination number and the magnitude and distribution of contact normal forces, and this dependence is macroscopically reflected as the influence of confining pressure and void ratio. The increase of coordination number and the uniformity of contact normal forces distribution with increasing confining pressure results in the stress exponent $n$ for elastic modulus being higher than 1/3 as predicted by the Hertz-Mindlin law. Furthermore, the simulations show that Poisson’s ratio of a granular packing is not a constant as commonly assumed, but rather it decreases as confining pressure increases. The variation of Poisson’s ratio appears to be a consequence of the increase of the coordination number rather than the increase of contact normal forces with confining pressure.     

Two-dimensional discrete element modelling of bender element tests on an idealised granular material

The small-strain (elastic) shear stiffness of soil is an important parameter in geotechnics. It is required as an input parameter to predict deformations and to carry out site response analysis to predict levels of shaking during earthquakes. Bender element testing is often used in experimental soil mechanics to determine the shear (S-) wave velocity in a given soil and hence the shear stiffness. In a bender element test a small perturbation is input at a point source and the propagation of the perturbation through the system is measured at a single measurement point. The mechanics and dynamics of the system response are non-trivial, complicating interpretation of the measured signal. This paper presents the results of a series of discrete element method (DEM) simulations of bender element tests on a simple, idealised granular material. DEM simulations provide the opportunity to study the mechanics of this testing approach in detail. The DEM model is shown to be capable of capturing features of the system response that had previously been identified in continuum-type analyses of the system. The propagation of the wave through the sample can be monitored at the particle-scale in the DEM simulation. In particular, the particle velocity data indicated the migration of a central S-wave accompanied by P-waves moving along the sides of the sample. The elastic stiffness of the system was compared with the stiffness calculated using different approaches to interpreting bender element test data. An approach based upon direct decomposition of the signal using a fast-Fourier transform yielded the most accurate results.     

Shear band formation in lunar regolith by discrete element analyses

Few studies in shear band formation have considered the environmental conditions on the Moon, which however are significant for lunar regolith failure in future lunar exploration activities. This paper presents a numerical investigation into the mechanical behavior and strain localization of lunar regolith by means of the discrete element method (DEM). A micromechanical contact model for lunar regolith accounting for van der Waals forces and rolling resistance has been developed, then implemented into a DEM code, PFC2D, and finally applied to analyze the strain localization of lunar regolith through biaxial tests. Biaxial tests without considering van der Waals force effect were also performed as reference to compare with. The distributions inside the sample of grid deformation, void ratio, velocity, averaged pure rotation rate (APR), force chains and local stress during shear banding are analyzed. The simulations show that persistent bands are differently formed under Moon and Earth conditions. Van der Waals forces and rolling resistance play crucial roles in choosing persistent bands from various transient micro-bands before the peak state. Van der Waals forces lead to increased dilation and particle rotation, and enlarged “meso-voids” in force chain distributions within the persistent shear bands. The thickness (inclination to the horizontal) of shear band for the regolith under Moon condition is smaller (larger) than that for regolith under Earth condition.The fields of velocity and APR can reveal the finest heterogeneity in particle displacement (translation and rotation) in the form of transient micro-bands even at the very beginning of shear.     

A discrete element study of settlement in vibrated granular layers: role of contact loss and acceleration

This paper deals with the vibration of granular materials due to cyclic external excitation. It highlights the effect of the acceleration on the settlement speed and proves the existence of a relationship between settlement and loss of contacts in partially confined granular materials under vibration. The numerical simulations are carried out using the Molecular Dynamics method, where the discrete elements consist of polygonal grains. The data analyses are conducted based on multivariate autoregressive models to describe the settlement and permanent contacts number with respect to the number of loading cycles.     

Linear-frictional contact model for 3D discrete element simulations of granular systems

The linear-frictional contact model is the most commonly used contact mechanism for discrete element (DEM) simulations of granular materials. Linear springs with a frictional slider are used for modeling interactions in directions normal and tangential to the contact surface. Although the model is simple in two dimensions, its implementation in 3D faces certain subtle challenges, and the particle interactions that occur within a single time step require careful modeling with a robust algorithm. The paper details a three-dimensional algorithm that accounts for the changing direction of the tangential force within a time step, the transition from elastic to slip behavior within a time step, possible contact sliding during only part of a time step, and twirling and rotation of the tangential force during a time step. Without three of these adjustments, errors are introduced in the incremental stiffness of an assembly. Without the fourth adjustment, the resulting stress tensor is not only incorrect but it is also no longer a tensor. The algorithm also computes the work increments during a time step, both elastic and dissipative.     

Simulation of granular crystallization of cubic particles in twisting container using discrete element method

Granular compaction is a process in which the volume fraction, or density, of the granular materials increases when an excitation is applied. A recent experiment reported that twisting a large number of cubic particles in a cylindrical container leads to an ordered and dense arrangement. This structure is similar to the crystal lattice formed in solidification process. In this article, this phenomenon is repeated by using discrete element method (DEM) simulation. Two different shaped containers are used and it is found that the rectangular angles between the sidewalls and the bottom,namely wall effect, plays a key role. In addition, gravitation is also a very important parameter in this process. The higher gravitation added, the faster crystallization process is achieved. On the contrary, shear force due to friction between particles may slow down this process.     

An algorithm to generate random dense arrangements for discrete element simulations of granular assemblies

A discrete element simulation of a mechanical problem involving granular materials begins with the definition of the geometry of the sample to be analyzed. Since the dynamic sample preparation methods typically used in the practice are very time-consuming, constructive algorithms are becoming increasingly popular. This paper introduces a novel constructive method for the preparation of random, isotropic assemblies of contacting circular discs with a user-defined grain size distribution. The proposed approach is compared with other currently applied sample preparation methods.     

A mixed finite element procedure of gradient Cosserat continuum for second-order computational homogenisation of granular materials

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu–Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts.     

Vibration induced flow in hoppers: continuum and DEM model approaches

A 2D and a 3D discrete element model (DEM) simulation of cohesive spherical particles are applied to assess the benefit of point source vibration to induce flow in wedge-shaped hoppers. The model is closely compared with a continuum model based on arch stability. A significant aspect of this study is the scaling of the continuum system to a discrete system of 500 particles in 2D and 2500 particles in 3D. This illustrates how such models can complement each other. The continuum model can cope with a full-scale industrial system, but is complex with significant assumptions. The discrete approach is relatively simple at the particle level with minimal assumptions but computationally demanding. The DEM model supports the basic conclusions of the continuum model. The vibration source must be located at the appropriate height above the outlet on the hopper to optimise its flow enhancement. Too low and stable arches can form above. Too high and it might not break the stable arches in the material below. The passive/active nature of the material during vibration and flow is also illustrated. The DEM model also shows that low frequency high amplitude vibration can enable flow through small orifices.     

Impact response of elasto-plastic granular and continuum media: A comparative study

A comparative study of the impact response of three-dimensional ordered granular sphere packings and continuum half-spaces made of elastic-perfectly plastic materials is conducted. Energy dissipation and plastic zone volume are characterized, and scaling laws with respect to material properties, size and loading variables are derived for both continuum and discrete (granular) systems. Due to stress concentration at contacts, energy dissipation in granular systems occurs at much smaller impact loads than in continuum systems. At higher impact loads, the fraction of energy dissipated and the extent of plastic zone are much larger in the discrete system than in the continuum case. Though the size of plastic zone is much larger in discrete systems, the volume of material involved in dissipating a fraction of impact energy is comparable for continuum and granular systems.